Monodromy Equivalence for Lam\'{e}-type Equations I: Finite-gap Structures and Cone Spherical Metrics

TL;DR

This paper explores the monodromy equivalence between classical and generalized Lamé equations, revealing finite-gap spectral structures and constructing cone spherical metrics with large conical singularities.

Contribution

It establishes a monodromy equivalence for Lamé-type equations, classifies spectral curves, and constructs cone spherical metrics with specified singularities.

Findings

Finite-gap structure of generalized Lamé equations derived.

Complete classification of spectral curves for certain parameters.

Construction of cone spherical metrics with large conical angles.

Abstract

Motivated by the finite-gap structure of the classical Lam\'{e} equation (1.2) and its central role in mathematical physics, generalized Lam\'{e}-type equations (1.12) are investigated. For the fundamental case $n=1$, a monodromy equivalence between the classical Lam\'{e} equation (1.18) and the generalized Lam\'{e}-type equation (1.19) is established. Two main applications are obtained: (i) the finite-gap structure of \ (1.19) is derived, together with a complete classification of the spectral curves $\sigma_{1}$ and $\sigma_{2}$ for $\tau\in i\mathbb{R}_{>0}$; and (ii) the monodromy equivalence is applied to the construction of cone spherical metrics with three large conical singularities, each with cone angle exceeding $2\pi$. A family of such metrics is shown to exhibits a blow-up configuration, which is described explicitly in terms of the monodromy data.

Equations

$$\zeta \left(z;\tau \right)$$

$$\sigma (z;\tau )$$

$$p\in E_{\tau }\backslash \left\{\frac{{\omega }_k}{2},k=0,1,2,3\right\}$$

$$y^{\prime \prime }(z)=\left[n(n+1)\wp \left(z\right)+\tilde{B}\right]y(z),\tilde{B}\in \mathbb{C},$$

$$y^{\prime \prime }(z)=q_n(z;p,T_1,T_2,B)y(z),$$

$$q_n(z;p,T,B)$$

$$+T_1(\zeta (z+p)-\zeta (z))+T_2(\zeta (z-p)-\zeta (z))+B$$

$$N_0=I{d}_{2\times 2}\text{ and }{N}_{\pm p}=-I{d}_{2\times 2}.$$

$$Y(z+{\omega }_i)=M_j(\mathbf{T};p)Y(z).$$

$$M_1{M}_2{M}_1^{-1}{M}_2^{-1}=N_p{N}_0{N}_{-p}=I{d}_{2\times 2}.$$

$$M_{1}(\mathbf{T};p)=\left(\begin{array}[c]{cc}e^{-2\pi is}&0\\ 0&e^{2\pi is}\end{array}\right),\text{ }M_{2}(\mathbf{T};p)=\left(\begin{matrix}e^{2\pi ir}&0\\ 0&e^{-2\pi ir}\end{matrix}\right),$$

$$M_{1}(\mathbf{T};p)=\pm\left(\begin{array}[c]{cc}1&0\\ 1&1\end{array}\right)\text{, }M_{2}(\mathbf{T};p)=\pm\left(\begin{matrix}1&0\\ \mathcal{D}&1\end{matrix}\right),\text{ }$$

$$M_{1}(\mathbf{T};p)=\pm\left(\begin{array}[c]{cc}1&0\\ 0&1\end{array}\right)\text{, }M_{2}(\mathbf{T};p)=\pm\left(\begin{matrix}1&0\\ 1&1\end{matrix}\right).$$

$$(T_1+T_2)\left(T_1-T_2+\frac{{\wp }^{\prime \prime }(p)}{2{\wp }^{\prime }(p)}\right)=0$$

$$B=\frac{T_1^2+T_2^2}{2}-\frac{T_1-T_2}{2}\zeta (2p)-\frac{3}{4}\wp (2p)-n(n+1)\wp (p).$$

$$AP(\tau ,p)=\left\{(T_1,T_2)|\text{(refAP condition 1) holds}\right\}$$

$$A{P}_0(\tau ,p)=\left\{(T_1,T_2)|T_1+T_2=0\right\},$$

$$A{P}_1(\tau ,p)=\left\{(T_1,T_2)|T_1-T_2+\frac{{\wp }^{\prime \prime }(p)}{2{\wp }^{\prime }(p)}=0\right\}.$$

$$AP(\tau ,p)=A{P}_0(\tau ,p)\oplus A{P}_1(\tau ,p)$$

$$A{P}_0(\tau ,p)\cap A{P}_1(\tau ,p)=\left\{\left(-\frac{{\wp }^{\prime \prime }(p)}{4{\wp }^{\prime }(p)},\frac{{\wp }^{\prime \prime }(p)}{4{\wp }^{\prime }(p)}\right)\right\}$$

$$\mathbf{T}=\left(A,-A\right),A\in \mathbb{C}.$$

$$y^{\prime\prime}(z)=\left[\begin{array}[c]{l}n(n+1)\wp(z)+\frac{3}{4}(\wp(z+p)+\wp(z-p))\\ +A(\zeta(z+p)-\zeta(z-p))+B\end{array}\right]y(z),\text{ }z\in E_{\tau},$$

$$B=A^2-\zeta (2p)A-\frac{3}{4}\wp (2p)-n(n+1)\wp (p).$$

$$\left(r(A),s(A)\right)\to -\left(r(A),s(A)\right)$$

$$\mathcal{D}(A)\to -\mathcal{D}(A)$$

$$\mathbf{T}=\left(T-\frac{{\wp }^{\prime \prime }(p)}{4{\wp }^{\prime }(p)},T+\frac{{\wp }^{\prime \prime }(p)}{4{\wp }^{\prime }(p)}\right),T\in \mathbb{C}.$$

$$y^{\prime\prime}(z)=\left[\begin{array}[c]{l}n(n+1)\wp(z)+\frac{3}{4}(\wp(z+p)+\wp(z-p))\\ +T(\zeta(z+p)+\zeta(z-p)-2\zeta(z))\\ -\frac{\wp^{\prime\prime}(p)}{4\wp^{\prime}(p)}(\zeta(z+p)-\zeta(z-p))+B\end{array}\right]y(z),\text{ }z\in E_{\tau},$$

$$B=T^2+\frac{{\wp }^{\prime \prime }(p)}{2{\wp }^{\prime }(p)}\zeta (p)-\frac{1}{2}(2n^2+2n-3)\wp (p).$$

$$\left(r(-T),s(-T)\right)=-\left(r(T),s(T)\right).$$

$$\mathcal{D}(-T)=-\mathcal{D}(T)\text{.}$$

$$A=-\frac{{\wp }^{\prime \prime }(p)}{4{\wp }^{\prime }(p)}.$$

Full text

1. 1 Introduction

2. 2 Finite-Gap Structure

3. 2.1 Finite-Gap Theory

4. 2.2 Deformation of Spectral Sets as p Varies

5. 3 Cone Spherical Metrics

6. 3.1 Existence of Cone Spherical Metrics

7. 3.2 Blow-up Analysis of the Family in Theorem 3.3

8. 3.3 Deformations of the Family in Theorem 3.3

9. 4 Spectral Theory

10. 5 Monodromy Theory and The Proofs

11. 5.1 Completely Reducible Case

12. 5.2 Non-completely Reducible Case

13. 5.3 The Proofs

1. Introduction

Let $\omega_{0}$ $=$ $0,$ $\omega_{1}$ $=$ $1,$ $\omega_{2}$ $=$ $\tau$, $\omega_{3}$ $=$ $1+\tau$, $\Lambda_{\tau}$ $=$ $\mathbb{Z}$ $\mathbb{\oplus}$ $\mathbb{\tau Z}$, and $E_{\tau}:=\mathbb{C}/\Lambda_{\tau}$ where $\tau$ $\in$ $\mathbb{H}$ $=$ $\left\{\tau|\operatorname{Im}\tau>0\right\}$. Let $\wp\left(z;\tau\right)$ be the Weierstrass elliptic function with periods $1$ and $\tau$. Let $\zeta\left(z;\tau\right)$ and $\sigma\left(z;\tau\right)$ be the Weierstrass zeta and sigma function, defined by

[TABLE]

which is entire on $\mathbb{C}$ and has a simple zero at $z=0$.

For brevity, we write $\wp\left(z;\tau\right)$, $\zeta\left(z;\tau\right)$, and $\sigma(z;\tau)$ simply as $\wp\left(z\right)$, $\zeta(z)$, and $\sigma(z)$, respectively. Let $p\in E_{\tau}$. Throughout this paper, we assume

[TABLE]

Let $n\in\mathbb{N}$. We consider two linear elliptic Fuchsian differential equations on elliptic curve $E_{\tau}$. The first is the classical Lamé equation [33, 60]:

[TABLE]

and the second is a generalized Lamé-type equation:

[TABLE]

where

[TABLE]

with parameters $T_{1},$ $T_{2},$ $B\in\mathbb{C}$, and $p\in E_{\tau}\backslash\{\frac{\omega_{k}}{2},k=0,1,2,3\}$. Equation (1.2) is known as classical Lamé equation in the literature, while (1.3) can be viewed as its natural variant, distinguished by the presence of nonzero residues at each singularity.

The monodromy problem for the integer Lamé equation (1.2) and its Treibich–Verdier generalizations, is considered fundamental due to its finite-gap structure or algebro-geometric structure. See [45, 58].

Finite-gap integration theory, also known as algebro-geometric integration, is a powerful method in the theory of integrable systems for constructing and analyzing solutions of certain nonlinear partial differential equations (PDEs), such as: the Korteweg-de Vries (KdV) equation [2, 17, 21, 22, 26, 27, 28, 29, 30, 31, 38, 50, 51], nonlinear Schrödinger equation (NLS) [2, 34, 35, 39, 40, 61], and the Sine-Gordon equation [37, 36]. This theory is deeply rooted in algebraic geometry, particularly the theory of compact Riemann surfaces and their Jacobians [47]. Beyond mathematics, it has found significant applications in physics, such as the stability analysis of critical droplets in bounded spatial domains [45, 46].

A central question motivating this work is whether the generalized Lamé-type equation (1.3) also admits a finite-gap structure, analogous to the classical Lamé equation (1.2). This series of studies develops a rigorous framework establishing the monodromy equivalence between the classical Lamé equation and the generalized Lamé-type equation in the punctured non-even case (see Case II below). This equivalence not only resolves the finite-band question but also yields new applications to spherical metrics with conical singularities.

The monodromy theory for the classical Lamé equation (1.2) had been thoroughly studied in [5, 44]. In this work, we focus our attention on the investigation of the monodromy theory associated with the generalized Lamé-type equation (1.3). In the present paper — constituting Part I of our study — we focus on the case $n=1$, which allows explicit computations and reveals the core ideas underlying the general theory.

A Lamé-type equation is called *apparent *if it is free of logarithmic singularity at each singularity. It is well known that the classical Lamé equation (1.2) is apparent for every $\tilde{B}\in\mathbb{C}$, due to the evenness of the potential $n(n+1)\wp\left(z;\tau\right)$ and the fact $z=0$ is a symmetric point on the elliptic curve $E_{\tau}$. In contrast, the generalized equation (1.3) involves nonzero residues and fails to be apparent unless certain explicit algebraic conditions are satisfied. See (1.6) below.

Denote by $AP(\tau,p)$ the space of apparent parameters. For a given parameter $\mathbf{T}$ $\mathbf{=}$ $\left(T_{1},T_{2}\right)$ $\in$ $AP(\tau,p)$, we define the monodromy representation of the generalized Lamé-type equation (1.3): Note that the local indices at $z=0$, $\pm p$ are $-n,$ $n+1$ and $-1/2,$ $3/2$, respectively, so the corresponding local monodromy matrices at $z=0$, $\pm p$ are

[TABLE]

We choose a base point $z_{0}\not\in\{0,\pm p\}$ such that $\{0,\pm p\}+\Lambda_{\tau}$ $\not\in$ $z_{0}+\mathbb{R}$ and $z_{0}+\tau\cdot\mathbb{R}$. Take a fundamental system of solution $Y(z;z_{0})$ of equation (1.3) in a small neighborhood $U_{z_{0}}$ of $z_{0}$. Define two global monodromy matrices $M_{1}(\mathbf{T};p)$ and $M_{2}(\mathbf{T};p)$ corresponding to the generators of the fundamental group $\pi_{1}(E_{\tau})$ via analytic continuation of $Y(z;z_{0})$ along loops associated with the periods $\omega_{1}=1$ and $\omega_{2}=\tau$, respectively:

[TABLE]

These matrices satisfy the monodromy relation

[TABLE]

Thus, the monodromy representation $\rho_{\tau,p}(\mathbf{T}):\pi_{1}(E_{\tau})\rightarrow SL(2,\mathbb{C})$, is abelian, leading to two cases:

(i) Completely Reducible case:* $M_{i}(\mathbf{T};p),$* $i$ $=$ $1,2$ can be simultaneously diagonalized as follows

[TABLE]

for some $r,s\in$ $\mathbb{C}$.

(ii) Non-Completely Reducible case: $M_{i}(\mathbf{T};p),$ $i$ $=$ $1,2,$ cannot be simultaneously diagonalized. Instead, they can be normalized to:

[TABLE]

where $\mathcal{D}$ $\in$ $\mathbb{C\cup\{\infty\}}$. In the case $\mathcal{D=\infty}$, it should be understood as

[TABLE]

The pair $\left(r,s\right)$ $\in$ $\mathbb{C}$ in case (i) and $\mathcal{D}$ $\in$ $\mathbb{C\cup\{\infty\}}$ in case (ii) , are respectively referred to as the monodromy data. A natural question is the following:

Question:* Let* $p$ $\in$ $E_{\tau}\backslash\{\frac{\omega_{k}}{2},k=0,1,2,3\}$. Given $\left(r,s\right)$ $\in$ $\mathbb{C}^{2}$ in case (i), and * $D$ $\in$ $\mathbb{C}\cup\{\infty\}$ in case (ii), does there exist a parameter $\mathbf{T}$* $\in$ $AP(\tau,p)$* such that the corresponding monodromy matrices $M_{i}(\mathbf{T},p),$ * $i$* $=$ $1,2$, are of the forms given in (1.4) and (1.5), respectively?*

To address the above question, we first examine the apparentness condition. The apparentness condition for the Lamé-type equation (1.3) imposes an explicit algebraic relation among the parameters $T_{1}$, $T_{2}$ and $B$. Specifically, the condition that all singularities are apparent is equivalent to the system:

[TABLE]

and $B$ is determined by

[TABLE]

Since the present paper deals with the case $n=1$, we will provide a direct proof of these relations via explicit computation in Section 4. The general case for arbitrary $n\in\mathbb{N}$ will be addressed in Part II of this series.

From (1.6), we have

[TABLE]

naturally decomposes into two components:

[TABLE]

[TABLE]

Clearly,

[TABLE]

and

[TABLE]

is the only singular point of $AP(\tau,p)$. Accordingly, we have the following two cases:

**Case I (Even symmetry): **If $\mathbf{T}$ $=$ $(T_{1},T_{2})$ $\in$ $AP_{0}(\tau,p)$, then we may set

[TABLE]

Under this assumption, the equation (1.3) becomes

[TABLE]

and by (1.7) $B$ is rephrased as follows

[TABLE]

It is easy to verify that, in this case, the potential is always an even elliptic function for any $A\in\mathbb{C}$. Since the potential is an even function, equation (1.8) is invariant under the transformation $z\rightarrow-z$. Under this transformation, the monodromy data change as follows:

• •

In completely reducible case (i):

[TABLE]

• •

In non-completely reducible case (ii):

[TABLE]

Accordingly, these monodromy data are regarded as equivalent under the symmetry $z\longmapsto-z$.

**Case II (Punctured Non-even symmetry): **If $\mathbf{T}$ $=$ $(T_{1},T_{2})$ $\in$ $AP_{1}(\tau,p)$, then we can write

[TABLE]

Thus, (1.3) becomes

[TABLE]

and

[TABLE]

It is clear that the potential remains elliptic, but—unlike in Case I—*it is not even unless $T=0$. *

For generalized Lamé-type equation (1.12), the transformation $z\longmapsto-z$ corresponds to $T\longmapsto-T$, since $B(T)$ $=$ $B(-T)$. As a result, the monodromy data of the equation with respect to $T$ and $-T$ relate as follows:

• •

In completely reducible case (i):

[TABLE]

• •

In non-completely reducible case (ii):

[TABLE]

When $T=0$, equation (1.12) yields to an even potential, equivalent to Case I (Even symmetry) with

[TABLE]

Notably, $T=0$, which corresponds to the intersection point

[TABLE]

is the unique point in $AP_{1}(\tau)$ for which equation (1.12) yields an even potential. This fact motivates the designation “Punctured Non-even Symmetry” for this case. A generalized Lamé-type equation (1.12) associated with $\mathbf{T}$ $\mathbf{\in}$ $AP_{1}(\tau,p)$ $\backslash$ $\{\mathbf{T}_{\ast}\}$ is called non-even symmetry.

The monodromy question above naturally divides into two distinct cases:

Case I. $\mathbf{T}$ $\mathbf{\in}$ $AP_{0}(\tau,p)$. The monodromy theory for potentials in this case has been extensively developed in [8, 9, 14], where deep connections were established between isomonodromic deformations of the family (1.8) and the elliptic form of the Painlevé VI equation (EPVI).

The EPVI associated to the equation (1.8) is given by

[TABLE]

with parameters

[TABLE]

**Theorem A. ([8] ) **The generalized Lamé-type equation in the even symmetry case (1.8) with parameters $\left(p(\tau),A(\tau)\right)$, preserves its monodromy as the moduli parameter $\tau$ varies if and only if $p(\tau)$ is a solution of EPVI (1.17 ).

According to the characterization of the monodromy matrices described in (i) and (ii), any solution $p^{(n)}(\tau)$ to EPVI (1.17) can be classified as follows:

(a) Completely reducible solution: $p(\tau)=p_{r,s}^{(n)}(\tau)$, where the associated equation (1.8) is completely reducible, and $\left(r,s\right)\in$* $\mathbb{C}^{2}$* is the monodromy data.

(b) Non-completely reducible solution: $p(\tau)=p_{\mathcal{D}}^{(n)}(\tau)$, where the associated equation (1.8) is non-completely reducible and $\mathcal{D}\in$* $\mathbb{C}\cup\{\infty\}$* is the monodromy data.

**Theorem B. *([8]) Given $\left(r,s\right)$ $\in$ $\mathbb{C}^{2}$ in case (i), or $\mathcal{D}$ $\in$ $\mathbb{C}\cup\{\infty\}$ in case (ii), there exists $\left(p,\text{ }A\right)$ such that the corresponding monodromy matrices $M_{i}(p,A),$ * $i$ $=$ $1,2$, are of the forms given in (1.4) or (1.5), respectively, if and only if $p=p_{r,s}^{(n)}(\tau)$ or $p=p_{\mathcal{D}}^{(n)}(\tau)$, evaluated at $\tau$.

Case II. $\mathbf{T}$ $\mathbf{\in}$ $AP_{1}(\tau,p)$. In this case, the analysis of the associated monodromy theory becomes considerably more intricate due to the non-vanishing residues at $\pm p$ and the lack of even symmetry in the potential if $T\not=0$.

The primary goal of the present paper is to address the monodromy question for $\mathbf{T}$ $\mathbf{\in}$ $AP_{1}(\tau,p)$ in the case $n$ $=$ $1$.

When $n=1$, the classical Lamé equation and the Lamé-type equation in Case II take the following forms respectively:

[TABLE]

and

[TABLE]

with

[TABLE]

Our main result asserts that, for any $p$ satisfying (1.1), the generalized Lamé-type equation (1.19)p parameter $T$ and $B$ given by (1.20) is monodromy equivalent to the classical Lamé equation (1.18) in the following sense:

Theorem 1.1**.**

Assume (1.1). Given $\left(r,s\right)$ $\in$ $\mathbb{C}^{2}$ in case (i) (1.4), or $\mathcal{D}$ $\in$ $\mathbb{C}\cup\{\infty\}$ in case (ii) (1.5), the following statements are equivalent:

(1) There exists $T$ $\in\mathbb{C}$, with $B$ given by (1.20) such that the monodromy matrices $M_{i}(p,T)$ $\in$ $SL(2,\mathbb{C}),$ for $i$ $=$ $1,2$, of (1.19)p correspond to the given $\left(r,s\right)$ or $\mathcal{D}$.

(2) There exists $\tilde{B}\in\mathbb{C}$ such that the monodromy matrices $M_{i}(\tilde{B})$ $\in$ $SL(2,\mathbb{C})$, for $i=1,2,$ of the classical Lamé equation (1.18) correspond to the same $\left(r,s\right)$ or $\mathcal{D}$.

Moreover, the correspondence between (1) and (2) is a two-to-one mapping

[TABLE]

given by the formula

[TABLE]

Remark 1.2**.**

Theorem 1.1 holds, in fact, for arbitrary $n\in\mathbb{N}$. A complete proof will be presented in Part II. In this paper, we focus on the case $n=1$, providing explicit formulas and computations that serve as the foundation for the general theory.

A direct corollary of Theorem 1.1 is stated as follows.

Corollary 1.3**.**

Suppose $\left(r,s\right)$ $\in$ $\mathbb{C}^{2}$ in case (i), or $\mathcal{D}$ $\in$ $\mathbb{C}\cup\{\infty\}$ in case (ii), represents the monodromy data of (1.18) for some $\tilde{B}$ $\in$ $\mathbb{C}$. For each $p$ satisfying (1.1), let the parameters $T(p)$ and $B(p)$ be given by (1.21) and (1.20), respectively. Then monodromy matrices

[TABLE]

of (1.19)p remain invariant and coincide with the corresponding monodromy matrices of (1.18) associated with $\tilde{B}$.

The next theorem emphasizes that the singular case $T(p)=0$ must occurs for some $p$ in Corollary 1.3.

Theorem 1.4**.**

Let $\tau\in\mathbb{H}$ and let $\left(T(p),B(p)\right)$ be as in Corollary 1.3, with (1.19)p completely reducible and having monodromy data $\left(r,s\right)$. Then

[TABLE]

where $p_{r,s}^{(1)}(\tau)$ denotes the solution of EPVI (1.17) with $n$ $=$ $1$, evaluated at $\tau$.

We now investigate the limiting behavior as $p\rightarrow\frac{\omega_{k}}{2}$. Although the singularity $p$ in the equation (1.19)p is not well-defined when

[TABLE]

we will show that the generalized Lamé-type equation (1.19)p with $\left(T(p),B(p)\right)$ chosen to preserve the monodromy matrices, converges uniformly on every compact subset of $E_{\tau}\backslash\{0\}$ as $p\rightarrow\frac{\omega_{k}}{2}$, for any $k=0,1,2,3.$

Moreover, the monodromy matrices

[TABLE]

associated with the limiting Lamé-type equation as $p\rightarrow\frac{\omega_{k}}{2},$ are determined by $M_{i}(p,T(p))$, which, by Corollary 1.3, are independent of $p$.

Theorem 1.5**.**

Let $\left(T(p),B(p)\right)$ be as in Corollary 1.3. Namely,

[TABLE]

(i) As $p\rightarrow 0$, the generalized Lamé-type equation in the non-even case (1.19)p, with parameters $\left(T(p),B(p)\right)$ converges uniformly to the classical Lamé equation (1.18) associated with $\tilde{B}$. The associated monodromy matrices $M_{i}^{(0)}(\tilde{B})$ of the limiting Lamé equation satisfy

[TABLE]

(ii) As $p\rightarrow\frac{\omega_{k}}{2}$, for any $k=1,2,3$, the generalized Lamé-type equation in the non-even case (1.19)p, with $\left(T(p),B(p)\right)$ converges uniformly to the Lamé-type equation

[TABLE]

where

[TABLE]

and

[TABLE]

Furthermore, the monodromy matrices $M_{i}^{(k)}(\tilde{T}_{k}),$ $i=1,2,$ associated with (1.22) satisfy

[TABLE]

where the signs are given by

[TABLE]

Organization of the Paper:

• •

Section 2. We extend the notion of finite-gap potentials to include the generalized Lamé-type equation (1.3), and for $n=1$, establish the finite-gap structure via Theorem 1.1 and describe the spectral arcs for $\tau\in i\mathbb{R}_{>0}$.

• •

Section 3. Applications to cone spherical metrics with three large odd conical singularities are presented. A family of such metrics exhibiting a blow-up configuration is analyzed via the associated multiple Green function, and the configuration is described explicitly in terms of the corresponding monodromy data.

• •

Section 4. We introduce spectral polynomials and Baker-Akhiezer functions, adapted from KdV theory, and employ them to study the monodromy representations for Case II (punctured non-even case).

• •

Section 5. We resolves the monodromy problem outlined above and gives the complete proof of Theorem 1.1, Theorem 1.4 and Theorem 1.5.

Acknowledgement: Ting-Jung Kuo was supported by NSTC 113-2628-M-003-001-MY4. He is also grateful to the National Center for Theoretical Sciences (NCTS) for its constant support.

2. Finite-Gap Structure

Finite-gap integration theory plays a central role in the analysis of nonlinear partial differential equations arising in mathematical physics. Motivated by the seminal works of Gesztesy and Weikard [30, 28, 29, 27, 31] on the KdV and AKNS hierarchies; Gesztesy, Holden, Michor, and Teschl [24, 25] on the Camassa–Holm and Ablowitz–Ladik hierarchies; and Gesztesy, Unterkofler, and Weikard [26] on Calogero–Moser systems, we extend the concept of finite-gap potentials to encompass the generalized Lamé-type equation in the punctured non-even case (1.19)p and establish its finite-gap property. In addition, we describe the spectral arcs for the case $\tau\in i\mathbb{R}_{>0}$.

Throughout this section, we denote the Case II potential in equation (1.19)p by $q(z;T)$, where $B$ is determined by (1.20).

2.1. Finite-Gap Theory

Let $T\in\mathbb{C}$. Consider a nonconstant elliptic function $h(z;T)$ in the variable $z$ with periods $1$ and $\tau$, subject to the following two assumptions:

• (1)

For fixed $z$, $h(z;\cdot)$ is meromorphic on $\mathbb{C}$;

• (2)

The second-order differential equation

[TABLE]

is assumed to be an apparent Fuchsian equation, that is, all of its singularities are regular and the local solutions are single-valued (i.e., free of logarithmic terms).

Fix $z_{0}\in\mathbb{C}$ such that $h(x+z_{0};T)$ is smooth on both $\mathbb{R}$ and $\tau\cdot\mathbb{R}$. Consider the following Hill’s equation

[TABLE]

Definition 2.1**.**

Equation (2.2) is said to be conditionally stable if it admits a nontrivial bounded solution on $\mathbb{R}$ (or $\tau\cdot\mathbb{R}$). The associated spectral sets are defined as

[TABLE]

that is, the sets of parameters $T$ for which (2.2) is conditionally stable on the respective domains.

Let $M_{j}(T)$ denote the monodromy matrix of equation (2.1) with respect to the shift $z\rightarrow z+\omega_{j}$. The Hill discriminant of (2.1) along the $\omega_{j}$ direction is defined by

[TABLE]

Note that $\Delta_{j}(T)$ is independent of the choice of fundamental system of solutions. By the classical Floquet theory, $\sigma_{j}$ can be characterized as

[TABLE]

Definition 2.2**.**

A point $T_{0}\in\sigma_{j}$ is called an endpoint if an odd number of semiarcs of $\sigma_{j}$ meet at $T_{0}$.

For any $T\in\mathbb{C}$, set

[TABLE]

The following lemma shows that $d_{j}(T)$ completely characterizes the finite endpoints of $\sigma_{j}$.

Lemma 2.3**.**

$T_{0}\in\mathbb{C}$* is an endpoint of $\sigma_{j}$ if and only if $d_{j}(T_{0})$ is odd.*

Proof.

Suppose $T_{0}\in\mathbb{C}$ is an endpoint of $\sigma_{j}$. Without the loss of generality, assume $\Delta_{j}(T_{0})=1$. Then locally,

[TABLE]

Hence by (2.5), we obtain

[TABLE]

which is odd.

Conversely, assume $d_{j}(T_{0})=k$ is odd. Then $\Delta_{j}(T_{0})=\pm 1,$ and near $T_{0}$,

[TABLE]

On the other hand,

[TABLE]

which implies

[TABLE]

Comparing (2.6) and (2.7), we conclude that $k^{\prime}=k$ and $l^{\prime}=\frac{l}{2}$. Thus, exactly $k$ semiarcs of $\sigma_{j}$ meet at $T_{0}$ and hence $T_{0}$ is an endpoint of $\sigma_{j}$. ∎

Definition 2.4** (finite-gap potential).**

The potential $h(z;T)$ is called finite-gap, if for each j=1,2, the spectral set $\sigma_{j}$ consists of finitely many bounded analytic arcs and finitely many semi-unbounded analytic arcs whose infinite endpoints tend to $\infty$.

Example 1**.**

[47, 58, 59, 60]** Let $n_{k}\in\mathbb{N}$, $k\in\{0,1,2,3\}$ with $\underset{0\leqslant k\leqslant 3}{max}n_{k}\geqslant 1$. The Darboux-Treibich-Verdier potential $\sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\frac{\omega_{k}}{2};\tau)+T$ is a finite-gap potential in the sense of Definition 2.4. In particular, the classical Lamé potential $n(n+1)\wp(z;\tau)+T$ with $n\in\mathbb{Z}_{>0}$ is a finite-gap potential in the sense of Definition 2.4.

In this section, we apply Theorem 1.1 to establish the finite-gap property of the potential $q(z;T)$ for the generalized Lamé-type equation (1.19)p. The main result is as follows.

Theorem 2.5**.**

Let $\tau\in\mathbb{H}.$ Suppose $p$ satisfies (1.1). Then $q(z;T)$ is a finite-gap potential in the sense of Definition 2.4. More precisely, for each $j\in\{1,2\}$, the following statements hold.

• (i)

The spectral set $\sigma_{j}$ admits the decomposition

[TABLE]

where

• •

each $\sigma_{j,k}$ is a bounded simple analytic arc;

• •

each $\sigma_{j,k,\infty}$ is a semi-infinite simple analytic arc tending to $\infty$;

• •

the set $\sigma_{j}$ is summetric with respect to the origin: $\sigma_{j}=-\sigma_{j}$.

• (ii)

All finite endpoints of $\sigma_{j}$ are given by the zeros of $\mathit{spectral\ polynomial}$ $Q(T)$ $(\text{The explicit expression of }Q(T)\text{ is given in }\eqref{spectral polynomial in noneven case})$. In particular,

[TABLE]

Here $T=0$ might be a double zero of $Q(T)$ in the degenerate case.

We first recall the spectral theory of classical Lamé equation. Consider the following Hill’s equation

[TABLE]

where $\wp(x+x_{0})$ is continuous on both $\mathbb{R}$ and $\tau\mathbb{R}$. Let $L(x)$ be a fundamental system of solution of (2.8). Then also $L(x+\omega_{j})$ is a fundamental system, so there exists a monodromy matrix $M_{j}(\tilde{B})$ with

[TABLE]

The Hill discriminant for each $j$ of (2.8) is defined as

[TABLE]

which is an entire function in $\tilde{B}$. The asssociated $\mathit{spectral\ set}$ of (2.8) is then

[TABLE]

It is known that the Lamé potential $2\wp(z)+\tilde{B}$ is an $\mathit{algebro}$-$\mathit{geometric}$ $\mathit{finite}$-$\mathit{gap}$ potential, equivalently a solution of the stationary KdV hierarchy. Its spectral polynomial $Q(\tilde{B})$ is

[TABLE]

The following fundamental result was proved in [30].

Theorem C.* ([30])* The spectral set $\tilde{\sigma}_{j}$ consists of one bounded spectral arc $\tilde{\sigma}_{j,1}$ and one semi-infinite arc $\tilde{\sigma}_{j,\infty}$ which tends to $-\infty+\langle q_{j}\rangle$, with

[TABLE]

That is,

[TABLE]

and the finite endpoints of these arcs coincide precisely with the zeros of the spectral polynomial $Q(\tilde{B})$.

We now turn to the investigation of the finite-gap property of the generalized Lamé-type potential $q(z;T)$.

Lemma 2.6**.**

For potential $q(z;T)$, the Hill discriminant satisfies $\Delta_{j}(T)=\Delta_{j}(-T)$. Consequently, $\sigma_{j}=-\sigma_{j}$ is symmetric with respect to the origin.

Proof.

Firstly we observe that $q(-z;T)=q(z;-T)$. Let $F_{0}(z)$ be a fundamental system of solution to equation $y^{\prime\prime}(z)=q(z;T)y(z)$ such that $F_{0}(z+\omega_{j})=M_{j}(T)F_{0}(z)$, then $F_{1}(z):=F_{0}(-z)$ solves $y^{\prime\prime}(z)=q(z;-T)y(z)$ and satisfies $F_{1}(z+\omega_{j})=M_{j}(T)^{-1}F_{1}(z)$. Since $M_{j}(T)\in\mathrm{SL}(2,\mathbb{C})$, then we have

[TABLE]

∎

Recall the definition of $d_{j}(T)$ from (2.5) and define

[TABLE]

respectively. The following lemma establishes the relationship between $d_{j}(T)$ and $\tilde{d}_{j}(\tilde{B})$.

Lemma 2.7**.**

Adapt notations above. Assume $T$ and $\tilde{B}\in\mathbb{C}$ satisfy (1.21). Then

[TABLE]

Proof.

By Theorem 1.1, we have

[TABLE]

Let $\tilde{B}_{0}=T_{0}^{2}-2\wp(p)$. Consider the local behavior of $\tilde{\Delta}_{j}$ at $\tilde{B}_{0}$:

[TABLE]

Using (2.11), we obtain

[TABLE]

which implies (2.10) holds. ∎

Recall the spectral polynomial $Q(T)$ in (4.40):

[TABLE]

The discriminant of $Q(T)$ is given by

[TABLE]

Thus, $Q(T)$ admits a multiple root at $T=0$ if and only if $\wp(p)=-\frac{e_{k}}{2}$ for some $k\in\{1,2,3\}$. These observations lead directly to the following result on the endpoints of $\sigma_{j}$.

Theorem 2.8**.**

All endpoints of $\sigma_{j}$ are zeros of $Q(T)$. In particular,

(i) If $Q(0)\not=0$, then

[TABLE]

(ii) If $Q(0)=0$, then

[TABLE]

Moreover, every endpoint $T$ of $\sigma_{j}$ satisfies $d_{j}(T)=1$.

Proof.

From Theorem 1.1, we have

Suppose $T_{0}$ is an endpoint of $\sigma_{j}$. We claim that $\tilde{B}_{0}=T_{0}^{2}-2\wp(p)$ is also an endpoint of $\tilde{\sigma}_{j}$. By Lemma 2.3 and Lemma 2.7, we have $d_{j}(T_{0})$ is an odd integer and

[TABLE]

Hence, Lemma 2.3 implies that $\tilde{B}_{0}$ is an endpoint of $\tilde{\sigma}_{j}$. By Theorem C, it follows that $\tilde{B}_{0}\in\{e_{1},e_{2},e_{3}\}$, which is equivalent to

[TABLE]

Therefore, $Q(T_{0})=0$ by (4.40).

Conversely, suppose $T$ is a zero of $Q(T)$. Then $\tilde{B}=T^{2}-2\wp(p)$ satisfies $\tilde{Q}(\tilde{B})=0$. By Theorem C, we have $\tilde{d}_{j}(\tilde{B})=1$.

(i) If $T\neq 0$, then Lemma 2.7 gives

[TABLE]

so by Lemma 2.3, $T$ is an endpoint of $\sigma_{j}$.

(ii)If $T=0$, then Lemma 2.7 implies

[TABLE]

which is even, and hence $T$ is not an endpoint of $\sigma_{j}$ by Lemma 2.3.

This completes the proof. ∎

Proof of Theorem 2.5.

Assume $T$ and $\tilde{B}$ are related by (1.21). Combining this with (2.4) and (2.9), we obtain

[TABLE]

Therefore, Theorem 2.5 follows directly from Theorem C , Lemma 2.6 and Theorem 2.8. ∎

2.2. Deformation of Spectral Sets as p Varies

Let $\tau\in i\mathbb{R}_{>0}$. Assume

[TABLE]

which is equivalent to requiring$\wp(p)\in(-\infty,+\infty)$. In this subsection, we provide a complete description of the spectral sets $\sigma_{1}$ and $\sigma_{2}$ for generic $p$. In particular, we prove that, as $\wp(p)$ varies along the real axis, $\sigma_{j}$ exhibits exactly seven distinct types of graphs. The main theorems are stated below.

Theorem 2.9**.**

Fix $\tau\in i\mathbb{R}_{>0}$. Suppose $\wp(p)\notin\mathbb{R}$. Then the spectral sets $\sigma_{j}$ decompose as

[TABLE]

where each $\sigma_{j,k}$ is a bounded simple analytic arc and each $\sigma_{j,k,\infty}$ is a semi-infinite simple analytic arc. Moreover, the following properties hold:

• •

For $j=1,2,$ the spectral sets $\sigma_{j}$ do not intersect either the real or the imaginary axis, namely,

[TABLE]

• •

Symmetry: $\sigma_{j,1}=-\sigma_{j,2}$, $\sigma_{j,1,\infty}=-\sigma_{j,2,\infty}$;

• •

Asymptotics: the infinite endpoints of $\sigma_{1,k,\infty}$ tend to $\pm i\infty$, while the infinite endpoints of $\sigma_{2,k,\infty}$ tend to $\pm\infty$, respectively.

More precisely, let $\sigma_{j}=\sigma_{j}^{\prime}\cup-\sigma_{j}^{\prime}$, where $\mathrm{Im}\sigma_{j}^{\prime}>0$. Then

• (1)

If $\mathrm{Im}\wp(p)>0$, then we have $\mathrm{Re}\sigma_{j}^{\prime}>0$;

• (2)

If $\mathrm{Im}\wp(p)<0$, then we have $\mathrm{Re}\sigma_{j}^{\prime}<0$.

Proof.

Lemma 2.6 implies that $\sigma_{j}=-\sigma_{j}$. Since $\wp(p)\notin\mathbb{R}$, then $\wp(p)\neq-\frac{e_{j}}{2}$, which implies $Q(0)\neq 0$. By Theorem 2.8, the finite endpoints of $\sigma_{j}$ coincide with the zeros of $Q(T)$, namely the set $\{\pm\sqrt{2\wp(p)+e_{j}},j=1,2,3\}$, and satisfy

[TABLE]

On the other hand, from Theorem D and (2.12), we have

[TABLE]

and

[TABLE]

Then we have

[TABLE]

and

[TABLE]

respectively. From the above discussions, we conclude that

• •

$\sigma_{j}$ consists of two bounded spectral arcs $\sigma_{j,k}$ and two infinte spectral arcs $\sigma_{j,k,\infty}$, $k=1,2$;

• •

The infinite endpoints of $\sigma_{1,k,\infty}$ tend to $\pm i\infty$, and the infinite endpoints of $\sigma_{2,k,\infty}$ tend to $\pm\infty$;

• •

$\sigma_{j,1}=-\sigma_{j,2}$ and $\sigma_{j,1,\infty}=-\sigma_{j,2,\infty}$.

Now assume $\mathrm{Im}\,\wp(p)>0$ and let $\sigma_{j}=\sigma_{j}^{\prime}\cup-\sigma_{j}^{\prime}$, where $\mathrm{Im}\,\sigma_{j}^{\prime}>0$. Then $\mathrm{Im}\,(2\wp(p)+e_{k})>0$. By Theorem 2.8, any finite endpoint $T$ of $\sigma_{j}^{\prime}$ satisfies

[TABLE]

which implies $\mathrm{Re}\,T>0$. Since $\sigma_{j}^{\prime}\cap\mathbb{R}=\emptyset=\sigma_{j}^{\prime}\cap i\mathbb{R}$, we conclude that $\mathrm{Re}\,\sigma_{j}^{\prime}>0$. The same arguments applies to the case $\mathrm{Im}\wp(p)<0$. This completes the proof. ∎

Theorem 2.10**.**

Fix $\tau\in i\mathbb{R}_{>0}$ and let $\wp(p)\in\mathbb{R}$. Then the spectral arcs $\sigma_{j}$ lie in $\mathbb{R}\cup i\mathbb{R}$ and satisfy

[TABLE]

Moreover, the following statements hold.

If $\wp(p)\in\left(-\infty,-\frac{e_{1}(\tau)}{2}\right)$, then

[TABLE]

If $\wp(p)=-\frac{e_{1}(\tau)}{2}$, then

[TABLE]

If $\wp(p)\in\left(-\frac{e_{1}(\tau)}{2},-\frac{e_{3}(\tau)}{2}\right)$, then

[TABLE]

If $\wp(p)=-\frac{e_{3}(\tau)}{2}$, then

[TABLE]

If $\wp(p)\in\left(-\frac{e_{3}(\tau)}{2},-\frac{e_{2}(\tau)}{2}\right)$, then

[TABLE]

If $\wp(p)=-\frac{e_{2}(\tau)}{2}$, then

[TABLE]

If $\wp(p)\in\left(-\frac{e_{2}(\tau)}{2},+\infty\right)$, then

[TABLE]

It is known that

[TABLE]

Recall the following important conclusion concerning the spectral sets of Lamé equation (2.8).

Theorem D. [7] Assume $\tau\in i\mathbb{R}_{>0}$. Then the spectral sets $\tilde{\sigma}_{j}$ of Lamé equation (2.8) satisfy

• (i)

$\tilde{\sigma}_{1}=(-\infty,e_{2}(\tau)]\cup[e_{3}(\tau),e_{1}(\tau)]$;

• (ii)

$\tilde{\sigma}_{2}=[e_{2}(\tau),e_{3}(\tau)]\cup[e_{1}(\tau),+\infty)$.

Consequently, the monodromy representation of classical Lamé equation (1.18) cannot be unitary for any $\tilde{B}\in\mathbb{C}$.

Proof of Theorem 2.10.

Since $\wp(p)\in\mathbb{R}$, any root of $Q(T)$ is either real or purely imaginary. By Theorem C, we see that

[TABLE]

Then (2.12) implies

[TABLE]

Analogous to the proof of Theorem 2.9, we also have (2.13) and (2.14) hold. Together with Lemma 2.6, we conclude that

• •

$\sigma_{j}$ is symmetric with respect to $\mathbb{R}$ and $i\mathbb{R}$;

• •

The infinite spectral arcs of $\sigma_{1}\ (\sigma_{2},\text{ respectively})$ tend to $\pm i\infty\ (\pm\infty)$.

Then for each case, it suffices to compute $d_{j}$ at endpoints.

• (1)

It follows from $\wp(p)\in\left(-\infty,-\frac{e_{1}(\tau)}{2}\right)$ that all roots of $Q(T)$ are simple and purely imaginary. Then we obtain from Theorem 2.8 that

[TABLE]

This proves Theorem 2.10 (1). The same argument applies to (3), (5) and (7).

• (2)

$\wp(p)=-\frac{e_{1}(\tau)}{2}$ implies $Q(0)=0$. From here and Theorem 2.8, we immediately obtain

[TABLE]

This proves Theorem 2.10 (2). The same argument applies to (4) and (6). This completes the proof.

∎

Corollary 2.11**.**

For $\tau\in i\mathbb{R}_{>0}$ , we have $\sigma_{1}\cap\sigma_{2}\setminus\{T|\,Q(T)=0\}=\emptyset$. Then the monodromy representation of the generalized Lamé-type equation (1.19)p cannot be unitary for any $T\in\mathbb{C}$.

Proof.

Note that the monodromy of the generalized Lamé-type equation (1.19)p is unitary if and only if monodromy data $(r,s)\in\mathbb{R}^{2}\setminus\frac{1}{2}\mathbb{Z}^{2}$, which is equivalent to

[TABLE]

Together Theorem 2.9 and Theorem 2.10, we conclude that $\sigma_{1}\cap\sigma_{2}\setminus\{T|\,Q(T)=0\}=\emptyset$. Therefore, the monodromy cannot be unitary for any $T\in\mathbb{C}$. ∎

3. Cone Spherical Metrics

3.1. Existence of Cone Spherical Metrics

Recently, cone spherical metrics have been studied extensively; see [5, 6, 7, 11, 3, 15, 12, 16, 18, 19, 20, 43, 44, 42] for significant contributions in this direction. Building upon the monodromy equivalence established in Theorem 1.1, we now apply these results to construct cone spherical metrics with prescribed conical singularities. We also examine related analytical aspects, highlighting how the integrable structure of the generalized Lamé-type equation naturally governs the geometry of such metrics.

The generalized Lamé-type equation (1.3) is closely related to the following curvature equation (PDE):

[TABLE]

which arises in the study of spherical metrics with conical singularities.

In conformal geometry, it corresponds to the problem of finding a conformal metric of constant positive curvature on the elliptic curve $E_{\tau}$, with conical singularities of cone angle $2\pi(2n+1)$ at [math], and and cone angle $4\pi$ at $\pm p$.

The approach to study the curvature equation (3.1) with an odd total angle relies on its integrability via Liouville Theorem, which states that any solution $u(z)$ of equation (3.1) is given as the form

[TABLE]

See [5, 49] for a proof. The function $f(z)$ is locally meromorphic and is commonly referred to as the developing map in the literature.

Through the representation (3.2), the problem of solving the curvature equation (3.1)p reduces to studying the generalized Lamé-type equation (1.3)p, with parameters, $\mathbf{T}$, $B$ satiafy the apparency condition, such that the corresponding monodromy matrices $M_{i}(\mathbf{T;}p),$ for $i$ $=$ $1,2,$ are unitary up to conjugation. The correspondence between the nonlinear curvature equation and the linear generalized Lamé-type equation—namely, the transition from PDE to ODE and vice versa—has become a well-established and widely used approach in the study of spherical metrics with conical singularities.

A notable special case is the curvature equation

[TABLE]

which is associated with the classical Lamé equation (1.2). The correspondence between these two equations has been established in detail in the seminal works [5, 44].

We briefly summarize the above discussion as follows.

**Theorem E. **Assume (1.1). Then the curvature equation associated with the singularity position $p$, denoted by (3.1)p admits a solution if and only if there exists parameters $\mathbf{T}$, $B\in\mathbb{C}$ such that the corresponding generalized Lamé-type equation (1.3)p is completely reducible, with associated monodromy data $\left(r,s\right)$ $\in$ $\mathbb{R}^{2}$ $\setminus$ $\frac{1}{2}\mathbb{Z}^{2}$.

Proof.

The proof of Theorem C. follows directly from the case of the model equation (3.3) and can be obtained by a straightforward modification; we therefore omit the details. See [5, 44, 41] for the proofs. ∎

It is also known that, if $u(z)$ of the form (3.2) is a solution, then the one-parameter family of functions

[TABLE]

also consists of solutions to the same equation. This family of solutions $\{u_{\beta}(z)|\beta$ $\in$ $\mathbb{R}\}$ exhibits blow-up behavior as described below:

• •

As $\beta\rightarrow+\infty$, it blows up at zeros of the developing map $f(z)$; while

• •

As $\beta\rightarrow-\infty$, it blows up at poles of the developing map $f(z)$.

This behavior is characteristic of bubbling solutions in geometric analysis, and is closely related to:

• •

The study of moduli spaces of flat connections or projective structures on Riemann surfaces;

• •

The analysis of holomorphic quadratic differentials and their associated monodromy representations;

• •

Mean field equations arising in statistical mechanics models and Chern–Simons–Higgs theory.

In particular, the parameter $\beta$ $\in$ $\mathbb{R}$ can be interpreted as a scaling parameter along a one-parameter family in the moduli space, where the geometry of the solution degenerates into concentrated curvature (delta-mass) configurations.

Suppose this family of solutions $\{u_{\beta}(z)|\beta$ $\in$ $\mathbb{R}\}$ contains at least one even solution. By scaling, we may assume that the solution corresponding to $\beta=0$, denoted $u_{0}(z)$, is even. In this case, it can be shown that $u_{0}(z)$ is the only even solution in the family.

Accordingly, we classify the family $\{u_{\beta}(z)|\beta$ $\in$ $\mathbb{R}\}$ as follows:

• •

If it contains an even solution (necessarily unique), it is called an even family of solutions.

• •

Otherwise, it is called a non-even family of solutions.

We remark that for the curvature equation (3.3), due to the presence of only a single conical singularity at the origin, every solution family $\{u_{\beta}(z)|\beta$ $\in$ $\mathbb{R}\}$ must necessarily be even.

It is straightforward to verify that

1. (1)

Every even family of solutions arises from the generalized Lamé-type equation (1.8) in the even symmetry case, for some $A$ and $B$ satisfying (1.9); while 2. (2)

Every non-even family of solutions corresponds to the generalized Lamé-type equation (1.12) in the non-even symmetry case, for some $T\not=0$ and $B$ satisfying (1.13).

Now, let us focus on the fundamental case $n=1$. In this setting, the two curvature equations take the form:

[TABLE]

and

[TABLE]

Recall (1.1) and define

[TABLE]

and

[TABLE]

Obviously, from the structure of the equation (3.6), the points $\pm p$ must be identified in both sets. Henceforth, we always treat $p$ and $-p$ as equivalent.

The main objective of this section is to characterize these two sets.

As a direct consequence of Theorems B and E, we obtain the following characterization for even families to the curvature equation (3.6)p.

**Theorem F.([14]) **For each $\tau\in\mathbb{H}$, the equation (3.6)p has an even family if and only if the singularity $p$ is given by

[TABLE]

where $p_{r,s}^{(1)}(\tau)$ denotes the solution of the Painlevé VI equation (1.17)n=1, evaluated at $\tau$, corresponding to $\left(r,s\right)$. Consequently,

[TABLE]

In fact, for any $\left(r,s\right)\in\mathbb{C}^{2}\setminus\frac{1}{2}\mathbb{Z}^{2}$, the solution $p_{r,s}^{(1)}(\tau)$ can be expressed explicitly. To this end, for such a pair $(r,s)$, we introduce the fundamental function

[TABLE]

which depends meromorphically on $(r,s)$. Since $(r,$ $s)$ $\not\in\frac{1}{2}\mathbb{Z}^{2}$, it is evident that $Z(r,s,\tau)$ $\not\equiv 0,\infty$ as a function of $\tau$, and is meromorphic in $\tau$.

This meromorphic function $Z(r,s,\tau)$ was initially introduced by Hecke in [32]. Hecke demonstrated that it is a modular form of weight one with respect to $\Gamma(n)$ whenever $(r,$ $s)$ is an $n$-torsion point. For this reason, $Z(r,s,\tau)$ is referred to as a premodular form.

The premodular form $Z(r,s,\tau)$ plays a central role in the monodromy problem for the classical Lamé equation (1.18) in completely reducible case, as stated below.

Theorem G.* ([43, 44]) Given any* $(r,s)$ $\in\mathbb{C}^{2}\setminus\frac{1}{2}\mathbb{Z}^{2}$. There exists $\tilde{B}\in\mathbb{C}$ such that the classical Lamé equation (1.18)τ is completely reducible and adimts monodromy data $\left(r,s\right)$* if and only if*

[TABLE]

Consequently, the curvature equation (3.5)τ has an even family if and only if there exists $(r,s)$ $\in\mathbb{R}^{2}\setminus\frac{1}{2}\mathbb{Z}^{2}$ such that (3.11) above holds.

We remark that more general premodular forms characterizing the monodromy problem in the completely reducible case have been constructed for (1.2) in [44], and for the Treibich–Verdier generalization in [10].

Let us denote

[TABLE]

Then the solution $p_{r,s}^{(1)}(\tau)$ can be explicitly expressed by

[TABLE]

By (3.10), we have

[TABLE]

Consequently, by (3.12) and the evenness of $\wp$, the solutions satisfy

[TABLE]

so $\pm p_{r^{\prime},s^{\prime}}^{(1)}(\tau)$ are identified as the same solution for any $\left(r^{\prime},s^{\prime}\right)\equiv\pm\left(r,s\right)$ mod $\mathbb{Z}^{2}$.

We now turn our attention to the study of non-even families of solutions to the curvature equation (3.6)p. Theorem E implies that finding such a family is equivalent to finding a parameter $T\not=0$, with $B$ determined by (1.13), such that the equation (1.19 )p has unitary monodromy, i.e., $(r,s)$ $\in$ $\mathbb{R}^{2}\setminus\frac{1}{2}\mathbb{Z}^{2}$.

Theorem 3.1**.**

Let $\tau\in\mathbb{H}$. The curvature equation (3.5) admits an even family if and only if there exists $p$ such that the curvature equation (3.6)p admits a non-even family.

Proof.

Suppose the curvature equation (3.5) admits an even family. There is $\tilde{B}\in\mathbb{C}$ such that the associated classical Lamé equation (1.18) is completely reducible with unitary monodromy. By the correspondecne (1.21), for any $p$ satisfying $\wp(p)\not=-\frac{1}{2}\tilde{B}$, the generalized Lamé-type equation with parameter $T$ determined by $T^{2}=B+2\wp(p)\not=0$ is completely reducible and admits the same unitary monodromy data. Consequently, the curvature equation (3.6)p admits a non-even family.

Converesly, suppose there exists $p$ such that (3.6)p admits a non-even family. By Theorem E, there is $T\not=0\in\mathbb{C}$ such that the generalized Lamé-type equation is completely reducible with unitary monodromy data. Applying Theorem 1.1 again, we conclude that the curvature equation (3.5) admits an even family. ∎

When $T=0$, the equation (1.19)p with unitary monodromy admits only an even family of solutions of the curvature (3.6)p. In this situation, by Theorem F, the singularity $p$ is given by $p=p_{r,s}^{(1)}(\tau),$ where $(r,s)$ $\in$ $\mathbb{R}^{2}\setminus\frac{1}{2}\mathbb{Z}^{2}$ is the monodromy data, which by (1.21), coincide with that of the classical Lamé equation (1.18) with parameter $\tilde{B}=-2\wp(p).$

Now, suppose the classical Lamé equation (1.18) with parameter $\tilde{B}$ has unitary monodromy data $(r,s)$ $\in$ $\mathbb{R}^{2}\setminus\frac{1}{2}\mathbb{Z}^{2}$. By Corollary 1.3, for each $p$, the parameter $T(p)$ of the generalized Lamé-type equation (1.19)p yielding the same unitary monodromy is determined by (1.21). By Theorem 1.4, when $p=p_{r,s}^{(1)}(\tau)$ (equivalently, when $-2\wp(p)=\tilde{B}$), we have $T(p)=0$. Thus, the equation equation (1.19)p reduces to the even symmetry case.

In view of the notation (1.16), we denote this special point by

[TABLE]

That is, $p_{\ast}$ is the unique point on $E_{\tau}$ such that the equation (1.19)${}_{p_{\ast}}$ with parameter $T=0$ shares the same monodromy data $\left(r,s\right)$ as the classical Lamé equation (1.18) with parameter $\tilde{B}$. In particular, for any $p\not=p_{r,s}^{(1)}(\tau)$, $T(p)\not=0$ and equation (1.19)p preserves exactly the same unitary monodromy as the classical Lamé equation (1.18) with parameter $\tilde{B}$.

Now, suppose the curvature equation (3.6)p admits a non-even family $\left\{u_{\beta}(z;p)\right\}$, associated with the generalized Lamé-type equation (1.19)p, which has parameter $T(p)$ and monodromy data $\left(r,s\right)$ $\in$ $\mathbb{R}^{2}\setminus\frac{1}{2}\mathbb{Z}^{2}$.

According to Theorem 1.1, the classical Lamé equation (1.18) with parameter

[TABLE]

also has the same monodromy data. Since both $\pm T(p)$ yield the same $\tilde{B}$, and the symmetry $z\longmapsto-z$ of the generalized Lamé-type equation (1.19)p is equivalent to $T\longmapsto-T$, it follows that $\left\{u_{\beta}(-z;p)\right\}$ also forms a non-even family of the curvature equation (3.6)p. Moreover, by (1.14), this second family corresponds to the monodromy data $\left(-r,-s\right)$ of (1.19)p with parameter $-T(p)$. This relationship is illustrated in the following diagram:

[TABLE]

Remark 3.2**.**

The equality $\left\{u_{\beta}(z;p)\right\}=\left\{u_{\beta}(-z;p)\right\}$ holds if and only if $T(p)=0$; in this case, the family is even.

In the sequel, we identify $\left\{u_{\beta}(z;p)\right\}$ with $\left\{u_{\beta}(-z;p)\right\}$, as they are connected via the transformation $z\rightarrow-z$. We denote this non-even family by

[TABLE]

where $\left(r,s\right)\in\mathbb{R}^{2}\setminus\frac{1}{2}\mathbb{Z}^{2}$ is the monodromy data of the associated generalized Lamé-type equation (1.19)p. With this convention, the preceding discussion yields the following theorem.

Theorem 3.3**.**

Let $\tau\in\mathbb{H}$. Suppose the curvature equation (3.5)τ admits an even family (necessarily unique) with monodromy data $(r,s)$ $\in$ $\mathbb{R}^{2}\setminus\frac{1}{2}\mathbb{Z}^{2}$. Then:

(i) For any $p\not=p_{r,s}^{(1)}(\tau)$, the curvature equation (3.6)p admits exact one non-even familiey $\left\{u_{\beta}(z;p)\right\}_{(r,s)}$, in the sense of (3.16), corresponding to $(r,s)$.

(ii) For $p=p_{\ast}:=p_{r,s}^{(1)}(\tau)$, the curvature equation (3.6)p admits exact one even family $\left\{u_{\beta}(z;p_{\ast})\right\}_{(r,s)}$, whose monodromy data is $(r,s)$.

By Theorem 3.1 and Theorem 3.3, for each $\tau\in\mathbb{H}$, the classification of solutions to the curvature equation (3.6)p with $p$ away from the symmetric points of the elliptic curve $E_{\tau}$, reduces to the analysis of the simpler curvature equation (3.5) with a single singular point.

The curvature equation (3.5) has been fully investigated in [43]. According to Theorem E, this analysis can be reduced to studying the associated equation (3.11) for $(r,s)$* $\in\mathbb{R}^{2}$* $\setminus\frac{1}{2}\mathbb{Z}^{2}$, which has already been carried out in [13]. We will review the relevant results below.

We first observe that the function $Z(r,s,\tau)$, as defined in (3.10), possesses certain modularity properties (see [13, (4.4)] or [41, (6.13)]) and satisfies (3.13 ). Consequently, it suffices to consider $Z(r,s,\tau)$ for $\left(r,s\right)$ $\in$ $\square$ and $\tau$ $\in$ $F_{0}$, where

[TABLE]

We further define

[TABLE]

Theorem H. ([13, Theorem 1.3.]) Let $\left(r,s\right)$ $\in$ $\square$. Then $Z(r,s,\tau)$ has a zero in $\tau\in F_{0}$ if and only if $\left(r,s\right)$ $\in$ $\Delta_{0}$. Moreover, for each such $\left(r,s\right)$, the zero $\tau$ $\in$ $F_{0}$ is unique.

Define

[TABLE]

By Theorem H, there exists a real-analytic map

[TABLE]

which is a bijection.

For $\left(r,s\right)\in\Delta_{0}$, let $\tau(r,s)\in\Omega$ denote the unique solution of $Z(r,s,\tau)=0$. In view of the notation in (3.15), we set

[TABLE]

Since $p_{\ast}(\tau(r,s))=p_{r,s}^{(1)}(\tau(r,s))$ satisfies the expression (3.12) and $Z(r,s,$ $\tau(r,s))$ $=$ [math], it follows that

[TABLE]

We thus obtain the following characterization of $\Lambda_{Noneven}^{(1)}$.

Theorem 3.4**.**

The set $\Lambda_{Noneven}^{(1)}$ can be characterized as follows:

(i) $\tau=\tau(r,s)\in\Omega$ for some $\left(r,s\right)\in\Delta_{0}$,

(ii) For such $\tau\left(r,s\right)\in\Omega$, the admissible singularity $p$ satisfies

[TABLE]

Equivalently,

[TABLE]

Furthermore, the exceptional point $p_{\ast}(\tau(r,s))$ is characterized by (3.17).

3.2. Blow-up Analysis of the Family in Theorem

3.3

Let $\tau=\tau(r,s)\in\Omega$. For each

[TABLE]

Theorem 3.3 ensures the existence of a unique non-even family of solutions to the curvature equation (3.6)p if $p\not=p_{\ast}(\tau(r,s))$; while it is an even family if $p$ $=$ $p_{\ast}(\tau(r,s))$. We denote the family obtained in Theorem 3.3 by $\left\{u_{\beta}(z;p)\right\}_{(r,s)}.$ As noted earlier, this family exhibits blow-up behavior as $\beta\rightarrow\pm\infty$, respectively. The natural problem is to determine the corresponding blow-up sets.

In this section, we investigate the blow-up sets associated with the non-even family of the curvature equation (3.6)p, obtained in Theorem 3.3. Specifically, we ask: for which $p$ does the curvature equation (3.6)p admit a family that blows up at the singularities $\pm p$? Equivalently, when can a blow-up configuration concentrate precisely at $p$ or $-p$?

The following theorem provides a definitive criterion for the family

$\left\{u_{\beta}(z;p)\right\}_{(r,s)}$ obtained in Theorem 3.3.

Theorem 3.5**.**

Let $\tau=\tau(r,s)\in\Omega$ and

[TABLE]

Then the unique family of solutions $\left\{u_{\beta}(z;p)\right\}_{(r,s)}$ of the curvature equation (3.6)p obtained in Theorem 3.3 blows up at the singularity $p$ as $\beta\rightarrow+\infty$ if and only if $p$ is determined by

[TABLE]

Proof.

According to Theorem 5.5 (ii), the assertion is equivalent to the existence of a point $p$ such that the corresponding

[TABLE]

Recall the relation

[TABLE]

By choosing $p$ such that

[TABLE]

we obtain

[TABLE]

and hence, equivalently,

[TABLE]

This proves the theorem. ∎

Next, we analyze the family of solutions of the curvature equation that blows up at the set $\{a_{1},a_{2}\}$, where $a_{j}\not=\pm p$. In this case, we have

[TABLE]

When the blow-up points do not coincide with the singularities $\pm p$, the blow-up analysis of the curvature equation (3.6)p, relies on the classical Pohozaev identity, which characterizes the blow-up points [6]. To make this more precise, we first introduce the Green function $G(z;\tau)$ on $E_{\tau}$, defined by

[TABLE]

where $\left|E_{\tau}\right|$ denotes the area of the torus. This function is even and has its only singularity at $z=0$. In the following, we omit the dependence on $\tau$ and simply write $G(z;\tau)=G(z)$.

Let $\eta$ be the linear map

[TABLE]

defined by

[TABLE]

Recall from [43] the following identity

[TABLE]

where

[TABLE]

It follows from (3.19) that for $\left(r,s\right)$ $\in$ $\{\left(1/2,0\right),$ $(0,1/2),$ $(1/2,1/2)\}$ mod $\mathbb{Z}^{2}$,

[TABLE]

for any $\tau\in\mathbb{H}$. In other words, each half-period $\frac{\omega_{k}}{2}$ is always a critical point of $G(z;\tau)$, and these are referred to as the trivial critical points. As shown in [43], for any torus $E_{\tau}$, the Green function $G(z;\tau)$ either has exactly three trivial critical points, or it additionally admits a pair of nontrivial critical points $\pm\sigma\in E_{\tau}$, where

[TABLE]

Theorem G further indicates that the presence of nontrivial critical points of $G(z;\tau)$ corresponds to flat tori characterized by the existence of even solutions to the curvature equation (3.5)τ. The pair of nontrivial critical points $\sigma$ and $-\sigma$ correspond to the blow-up points of the unique even family $\left\{v_{\beta}(z)|\beta\in\mathbb{R}\right\}$ of solutions, associated with the limits $\beta\rightarrow+\infty$ and $\beta\rightarrow-\infty$, respectively.

The blow-up analysis of the associated multiple Green function plays a crucial role in understanding cone spherical metrics. Very recently, Chen, Fu and Lin in [15] studied the curvature equation

[TABLE]

Since the total curvature of the curvature equation (3.20) is $8\pi$, any blow-up family of solutions admits only a single blow-up point. This allows the application of the method of anti-holomorphic dynamics developed by Bergweiler and Eremenko [3], together with Hitchin’s formula, to obtain refined results concerning the critical points of the associated multiple Green function

[TABLE]

In contrast, for the curvature equation (3.6), the total curvature is $16\pi$, and, generically, any blow-up family involves two blow-up points. Hence, the analysis becomes significantly more intricate.

For each $p\in E_{\tau(r,s)}\setminus\{\frac{\omega_{k}}{2},k=0,1,2,3\}$, we then introduce the associated multiple Green function, defined by

[TABLE]

This multiple Green function naturally arises in the analysis of blow-up configurations via the Pohozaev identity: its critical points characterize the locations of blow-up points whenever they are not equal to $\pm p$.

By differentiating and applying (3.19), we obtain that the pair $\{a_{1},a_{2}\}$ satisfies the following relations

[TABLE]

and

[TABLE]

where

[TABLE]

for a unique pair $\left(r,s\right)\in\square$. Moreover, by Theorem 3.5, we have

[TABLE]

The above equations (3.21) and (3.22) represent the critical point conditions for the multiple Green function $G_{p}(z_{1},z_{2})$.

To study the system of equations (3.21) and (3.22), we introduce the following notations: For $j=1,2,$ set

[TABLE]

By the classical relation for the Weierstrass elliptic function, each pair $\left(x_{j},y_{j}\right)$ lies on the elliptic curve and satisfies

[TABLE]

With this notation, equations (3.21) and (3.22) can be rewritten as the following linear system in $y_{1}$, $y_{2}:$

[TABLE]

together with the elliptic curve relations (3.24).

Hence, the existence of a blow-up family of solutions to the curvature equation (3.6)p blowing up at $\{a_{1},a_{2}\}$ subject to condition (3.18) as $\beta\rightarrow\pm\infty$ is equivalent to the existence of points $\left(x_{1},y_{1}\right)$ and $\left(x_{2},y_{2}\right)$ on the elliptic curve defined by (3.24) , such that $(y_{1},y_{2})$ forms a nontrivial solution of the linear system (3.25).

A straightforward computation shows that the determinant of the linear system (3.25) is given by

[TABLE]

The constraint (3.18) for $a_{1}$, $a_{2}$ implies that

[TABLE]

Thus, the determinant vanishes if and only if

[TABLE]

In this situation, the linear system (3.25) degenerates and reduces to

[TABLE]

Theorem 3.6**.**

Suppose the curvature equation (3.6)p admits a blow-up family $\{u_{\beta}(z)|\beta\in\mathbb{R}\}$, either even or non-even, of solutions blowing up at $\{a_{1},a_{2}\}$ subject to condition (3.18) as $\beta\rightarrow\pm\infty$. Adopt the notation (3.23) and set

[TABLE]

where $\left(r,s\right)$ is the unique real pair in $[0,1]\times[0,1]$ representing the monodromy data of the associated generalized Lamé-type equation (in either the even or punctured non-even symmetry case). Then:

(1)

[TABLE]

(2)

[TABLE]

Remark 3.7**.**

The blow-up family $\{u_{\beta}(z)|\beta\in\mathbb{R}\}$ in Theorem 3.6 is not necessary obtained from Theorem 3.3.

Theorem 3.6 and Theorem 3.3 can be used to characterize the blow-up points subject to condition (3.18). Suppose $\tau=\tau(r,s)\in\Omega$, that is, $Z(r,s,\tau(r,s))=0$. By Theorem 3.3, for each $p$, there exists a family of solutions

[TABLE]

This family of solutions blows up at $\{a_{1}^{\pm},a_{2}^{\pm}\}$ subject to condition (3.18) as $\beta\rightarrow\pm\infty$ whenever the singularity $p$ satisfies $2p$ $\not=$ $\pm\sigma$ (mod $\Lambda_{\tau}$). Moreover, this family $\{u_{\beta}(z;p)|\beta\in\mathbb{R}\}_{(r,s)}$ is a non-even family provided $p\not=p_{\ast}$.

By Theorem 3.6, we obtain

[TABLE]

Sincce $a_{1}\not=\sigma$, applying the addition formula for the Weierstrass function,

[TABLE]

equation (3.26) is equivalent to

[TABLE]

where $\left(x_{1},y_{1}\right)=\left(\wp(a_{1}),\wp^{\prime}(a_{1})\right)$.

Solving the system yields four solutions for $x_{1}$:

[TABLE]

where

[TABLE]

By symmetry in $x_{1}$ and $x_{2}$ in the equation (3.26), the blow-up configurations of the family $\{u_{\beta}(z;p)|\beta\in\mathbb{R}\}_{(r,s)}$ are determined by the monodromy data $\left(r,s\right)$ as follows:

• •

As $\beta\rightarrow+\infty$, the blow-up set $\{a_{1}^{+},a_{2}^{+}\}$ satisfies

[TABLE]

• •

As $\beta\rightarrow-\infty$, the blow-up set $\{a_{1}^{-},a_{2}^{-}\}$ satisfies

[TABLE]

Theorem 3.8**.**

Let $\tau=\tau(r,s)\in\Omega$, and $\{u_{\beta}(z;p)|\beta\in\mathbb{R}\}_{(r,s)}$ be the family of solutions obtained in Theorem 3.3. Suppose the singularity $p$ satisfies

[TABLE]

Then the blow-up set of $\{u_{\beta}(z;p)|\beta\in\mathbb{R}\}_{(r,s)}$ is characterized by (3.29) and (3.30) as $\beta\rightarrow\pm\infty$, respectively.

In particular, when $p=p_{\ast}$, by (3.17), the blow-up sets of the even family simplify to

[TABLE]

and

[TABLE]

The last statement (3.31) follows immediately by substituting $\wp(p_{\ast})=-\frac{1}{2}\wp(\sigma)$ into (3.29) and (3.30). Moreover, the point $\sigma$ $=$ $r+s\tau$, with $\left(r,s\right)\in$ $\mathbb{R}^{2}$ $\setminus\frac{1}{2}\mathbb{Z}^{2}$, also represents the blow-up point of the even family $\{v_{\beta}(z)|\beta\in\mathbb{R}\}_{(r,s)}$ of the curvature equation (3.5) as $\beta\rightarrow+\infty$ in Theorem 3.3.

Hence Theorem 3.8 establishes the correspondence between the blow-up set of $\{u_{\beta}(z;p)|\beta\in\mathbb{R}\}_{(r,s)}$ for the curvature equation (3.6)p and that of the curvature equation (3.5).

Let $\tau=\tau(r,s)\in\Omega$ and suppose $p$ satisfies

[TABLE]

According to Theorem 3.5, the family $\{u_{\beta}(z;p)|\beta\in\mathbb{R}\}_{(r,s)}$ blows up at the singularity $p$, removing a total curvature of $16\pi$, as $\beta\rightarrow+\infty$. By taking $2p\not=\pm\sigma$ and $2p\rightarrow\pm\sigma$ (mod $\Lambda_{\tau}$), we obtain

[TABLE]

which is equivalent to the vanishing of the discriminant

[TABLE]

Since $\Delta_{+}(p,\sigma)=0$, we obtain

[TABLE]

By Theorem 3.8, the opposite blow-up set of this family as $\beta\rightarrow-\infty$, can be characterized by

[TABLE]

We summarize the above discussion as below.

Corollary 3.9**.**

Let $\tau=\tau(r,s)\in\Omega$, and let

[TABLE]

be the family of solutions obtained in Theorem 3.3. Suppose $p$ satisfies

[TABLE]

Then:

(i) As $\beta\rightarrow+\infty$, $\{u_{\beta}(z;p)|\beta\in\mathbb{R}\}_{(r,s)}$ blows up at $p$, removing a total curvature of $16\pi.$

(ii) As $\beta\rightarrow-\infty$, $\{u_{\beta}(z;p)|\beta\in\mathbb{R}\}_{(r,s)}$ blows up at two distinct points $\{a_{1}^{-},a_{2}^{-}\}$, characterized by

[TABLE]

where each blow-up point removes a total curvature of $8\pi.$

The two distinct blow-up points $a_{1}^{-}$, $a_{2}^{-}$ may collapse to the singularity when $g_{2}(\tau)=0$. Indeed, consider

[TABLE]

It is well known from [43, Example 2.6] that

[TABLE]

That is,

[TABLE]

Suppose

[TABLE]

Then we have

[TABLE]

By $g_{2}(\rho)=0$, it follows that

[TABLE]

which implies

[TABLE]

Thus,

[TABLE]

Therefore, the family

[TABLE]

is an even family of solution to the curvature equation (3.6)${}_{\tau=\rho,p=p_{\ast}}$.

Moreover, $p_{\ast}$ satisfies

[TABLE]

It then follows from Theorem 3.5 and Corollary 3.9, $\{u_{\beta}(z;p_{\ast})|\beta\in\mathbb{R}\}_{\left(\frac{1}{3},\frac{1}{3}\right)}$ blows up at $p_{\ast}$ as $\beta\rightarrow+\infty,$ and at $-p_{\ast}$ as $\beta\rightarrow-\infty$.

Corollary 3.10**.**

Let $\tau=\rho=e^{\pi i/3}$. Then the even family

[TABLE]

to the curvature equation (3.6)${}_{\tau=\rho,p=p_{\ast}}$ blows up at $p_{\ast}$ as $\beta\rightarrow+\infty,$ and at $-p_{\ast}$ as $\beta\rightarrow-\infty$.

3.3. Deformations of the Family in Theorem 3.3

Fix moduli parameter $\tau=\tau(r,s)\in\Omega$. In this subsection, we investigate the deformation theory of the family $\{u_{\beta}(z;p)|\beta\in\mathbb{R}\}_{\left(r,s\right)}$ obtained in Theorem 3.3, in the following setting: Deform

[TABLE]

such that $p\rightarrow\frac{\omega_{k}}{2}$ for $k=0,1,2,3$.

Recall the correspondence (1.21):

[TABLE]

In general, we may assume the family $\{u_{\beta}(z;p)|\beta\in\mathbb{R}\}_{\left(r,s\right)}$ is non-even. We deform $p$ subject to the condition (3.32) such that $p\rightarrow\frac{\omega_{k}}{2}$, $k=0,1,2,3$. That is,

[TABLE]

(i) The case $k=0$.

By Theorem 1.5 (i), the generalized Lamé-type equation (1.19)p converges to the classical Lamé equation

[TABLE]

which is completely reducible and has the same monodromy data $\left(r,s\right)\in\Delta_{0}$. Consequently, the family $\{u_{\beta}(z;p)|\beta\in\mathbb{R}\}_{\left(r,s\right)}$ converges to the unique even family $\{v_{\beta}(z)|\beta\in\mathbb{R}\}_{\left(r,s\right)}$ of the curvature equation (3.5)τ=τ(r,s) as $p\rightarrow 0$ under condition (3.32).

(ii) The case $k=1,2,3.$

By Theorem 1.5 (ii), as $p\rightarrow\frac{\omega_{k}}{2}$ the generalized Lamé-type equation (1.19)p converges to the following Lamé-type equation

[TABLE]

where

[TABLE]

Since the generalized Lamé-type equation (1.19)p associated with the curvature equation (3.6)τ=τ(r,s),p is completely reducible, the limiting Lamé-type equation (3.33) must also be completely reducible. Denote the monodromy data of (3.33) by $\left(\tilde{r},\tilde{s}\right)\in\mathbb{C}^{2}\setminus\frac{1}{2}\mathbb{Z}^{2}$. Then, by (1.23), the relation between the monodromy data of (1.19)p and that of (3.33) is given by

[TABLE]

where

[TABLE]

With this notation (3.34), we obtain

[TABLE]

Since $Z(\tilde{r}_{k},\tilde{s}_{k},\tau(r,s))=0$, Theorem 6.1 in [41, Theorem 1.3] implies $\tilde{T}_{k}\not=0$. By Theorem 1.3 in [41], the Lamé-type equation (3.33) yields the unique non-even family

[TABLE]

of solutions to the curvature equation

[TABLE]

In particular, the family $\{u_{\beta}(z;p)|\beta\in\mathbb{R}\}_{\left(r,s\right)}$ converges to the unique non-even family $\left\{v_{k,\beta}(z)|\beta\in\mathbb{R}\right\}_{\left(\tilde{r},\tilde{s}\right)}$ of the curvature equation (3.35).

Theorem 3.11**.**

Let $\tau=\tau(r,s)\in\Omega$ be fixed and $\{u_{\beta}(z;p)|\beta\in\mathbb{R}\}_{\left(r,s\right)}$ be the family obtained in Theorem 3.3. Suppose $p\not=p_{\ast}\rightarrow\frac{\omega_{k}}{2}$, $k=0,1,2,3,$ subject condition (3.32). Then:

(i) For $k=0$, the non-even family $\{u_{\beta}(z;p)|\beta\in\mathbb{R}\}_{\left(r,s\right)}$ converges uniformly to the unique even family $\{v_{\beta}(z)|\beta\in\mathbb{R}\}_{\left(r,s\right)}$ of the curvature equation (3.5)τ=τ(r,s).

(ii) For $k=1,2,3$, the non-even family $\{u_{\beta}(z;p)|\beta\in\mathbb{R}\}_{\left(r,s\right)}$ converges uniformly to the unique non-even family $\left\{v_{k,\beta}(z)|\beta\in\mathbb{R}\right\}_{\left(\tilde{r},\tilde{s}\right)}$ of the curvature equation (3.35), where the monodromy data satisfy

[TABLE]

with $\left(\tilde{r}_{k},\tilde{s}_{k}\right)$ defined in (3.34).

Remark 3.12**.**

This theorem shows a dichotomy in the limiting behavior at half-periods. As $p\rightarrow 0$, the family reduces to the even branch of the curvature equation, preserving the monodromy data $\left(r,s\right)$. In contrast, as $p\rightarrow\frac{\omega_{k}}{2}$ ($k=1,2,3$) , the limiting families become non-even, with the monodromy data shifted by a half-lattice translation. Thus, the distinction between even and non-even families is governed by the lattice symmetries of the monodromy data under half-period shifts. This result generalizes Theorem 1.3 in [41].

4. Spectral Theory

Recall the assumption (1.1). In this section, we aim to establish the spectral theory for the generalized Lamé-type equation:

[TABLE]

where $T_{1},T_{2},B\in\mathbb{C}$, and the potential $q(z;T_{1},T_{2},B)$ is given by (1.3)n=1:

[TABLE]

Firstly, we apply the standard Frobenius method to derive the apparent condition for equation (4.1)p.

Lemma 4.1**.**

The generalized Lamé-type equation (4.1)p is apparent at the singularities $z=0,\pm p$ if and only if $T_{1},T_{2},B$ satisfy:

[TABLE]

where $B$ is determined by

[TABLE]

Proof.

The local exponents of (4.1) at the singularities $z=0,\,p,$ and $-p$ are

[TABLE]

Accordingly, the singularities at $z=0,\pm p$ are apparent precisely when there exists a local solution corresponding to the smaller exponent at each singularity—namely $-1$ at $z=0$ and $-\frac{1}{2}$ at $z=\pm p$. These solutions admit the following local expansions:

[TABLE]

Substituting the expansions in (4.5) into (4.1)p and matching coefficients yield recursive relations for the coefficients $c_{m}$ and $d_{m_{\pm p}}$. These recursion relations impose algebraic constraints on the parameters $T_{1},T_{2},B$, which must satisfy the conditions given in (4.3) and (4.4).

This completes the proof. ∎

Lemma 4.1 asserts that

[TABLE]

which naturally decomposes into two components $AP_{i}(\tau,p)$, $i=0,1$, as follows:

[TABLE]

where the corresponding potential $q(z;T_{1},T_{2},B)$ is

[TABLE]

with $B$ determined by

[TABLE]

[TABLE]

where the potential $q(z;T_{1},T_{2},B)$ is given by

[TABLE]

with $B$ determined by

[TABLE]

These two components intersect precisely at one singular point

[TABLE]

We now turn to the spectral analysis of (4.1)p. Since the arguments for the even-symmetry case (4.7) and for the punctured non-even symmetry case (4.10) are essentially identical — and our primary focus is the latter — we present full details only for the generalized Lamé-type equation in the punctured non-even case (1.19)p. The corresponding results for the even-symmetry case are provided in Theorem 4.5.

To proceed, we consider the second symmetric product of equation(1.19)p, which is a third-order Fuchsian equation:

[TABLE]

Theorem 4.2**.**

Up to a nonzero multiple, there exists a unique non-trivial elliptic solution $\Phi_{e}(z;T)$ of (4.14), given by

[TABLE]

Since the local exponents of (4.14)p at $z=0$ and $z=\pm p$ are, respectively, $-2,1,4$ and $-1,1,3$, we may assume $\Phi_{e}(z;T)$ as follows:

[TABLE]

for some $c_{0},c_{1},c_{2},c_{3}\in\mathbb{C}$. If $c_{0}=0$, the local exponent of $\Phi_{e}(z;T)$ at $z=0$ forces $c_{1}=c_{2}=c_{3}=0$; hence we may assume that $c_{0}=1$.

Define the elliptic function

[TABLE]

where $\Phi_{e}(z;T)$ is given by (4.16). Its Laurent expansion at $z=0$ is

[TABLE]

with coefficients

[TABLE]

Proof of Theorem 4.2.

With notations as above. Since $\Phi_{e}(z;T)$ solves (4.14) if and only if $G\equiv 0$, the coefficients in the Laurent expansion (4.18) must satisfy $d_{j}(T)=0$ for $-4\leq j\leq 0$.

From $d_{-4}(T)=0$ and $d_{-1}(T)=0$, we obtain

[TABLE]

Substituting (4.35) into $d_{-3}(T)=0$ yields

[TABLE]

Furthermore, by (4.35) and (4.36), $d_{-2}(T)=0$ and $d_{0}(T)$ hold automatically. Next, we prove that $G(z;T)$ is holomorphic at $z=\pm p$. A direct computation shows that

[TABLE]

By applying the addition formula

[TABLE]

we find that the residue in (4.37) vanishes. Consequently, the function $G$ is also holomorphic at $z=p$, and by symmetry, it is also holomorphic at $z=-p$.

This completes the proof. ∎

Define

[TABLE]

Since $\Phi_{e}(z;T)$ solves (4.14), a direct computation shows that $\frac{d}{dz}Q(z;T)\equiv 0$. Hence, $Q(z;T)$ is independent of $z$, and we may denote it simply by $Q(T)$. Moreover, as $q(z;T)$ and $\Phi_{e}(z;T)$ are polynomial in $T$, $Q(T)$ is then a polynomial in $T$. A direct computation shows that

[TABLE]

By analogy with the KdV hierarchy, we refer to$Q(T)$ as the spectral polynomial. The associated spectral curve is then defined by

[TABLE]

For $P=(T,C)\in\Gamma(\tau,p)$, we define its dual point by $P^{\ast}:=(T,-C)\in\Gamma(\tau,p)$.

The spectral polynomial and corresponding spectral curve for the classical Lamé equation (1.18) were derived in [45] and are given by

[TABLE]

with the associated spectral curve defined as

[TABLE]

Surprisingly, in light of (4.40) and (4.42), there exists a two-to-one correspondence between the two spectral curves:

[TABLE]

This correspondence reflects the monodromy equivalence between the classical Lamé equation (1.18) and the generalized Lamé-type equation (1.19)p as stated in Theorem 1.1.

Next, we define the Baker-Akhiezer function for each $P\in\Gamma(\tau,p)$ as follows: For any $P=(T(P),C(P))\in\Gamma(\tau,p)$, define the meromorphic function

[TABLE]

A direct computation shows that $\phi(P;z)$ satisfies the Riccati equation:

[TABLE]

Proposition 4.3**.**

Let $P=(T,C)\in\Gamma(\tau,p)$. Suppose $z_{0}$ is a pole of $\phi(P;z)$.

• (i)

If $z_{0}\in\{0,p,-p\}$, one has

[TABLE]

• (ii)

If $z_{0}\notin\{0,p,-p\}$, then

[TABLE]

Proof.

From (4.45), the poles of $\phi(P;z)$ are simple and arise either from the poles or from the zeros of $\Phi_{e}(z;T)$. If $z_{0}$ is a pole of $\Phi_{e}(z;T)$, i.e. $z_{0}\in\{0,\pm p\}$, then since

[TABLE]

it follows that

[TABLE]

If $\Phi_{e}(z;T)$ vanishes at some point $z_{0}\notin\{0,\pm p\}$, then $z_{0}$ must be a simple zero of $\Phi_{e}(z;T)$. Substituting into the Riccati equation (4.46) yields $\operatorname{Res}_{z=z_{0}}\phi(P;z)=1$. This completes the proof. ∎

Given a base point $z_{0}\notin\{0,\pm p\}$, we define the Baker-Akhiezer function at $P\in\Gamma(\tau,p)$ by

[TABLE]

where the integration path is chosen to avoid the singularities of the meromorphic function $\phi(P;\xi)$. By Proposition 4.3, $\psi(P;z,z_{0})$ is a multi-valued meromorphic function whose analytic continuation around $\pm p$ produces nontrivial monodromy, reflecting the presence of branch points at $\pm p$.

By the Riccati equation (4.46), it is easy to verity that $\psi(P;z,z_{0})$ solves the generalized Lamé-type equation (1.19)p with parameter $T(P)$, where $T=T(P)$ is the $T$-coordinate of $P\in\Gamma(\tau,p)$. Thus, each point $P\in\Gamma(\tau,p)$* corresponds to an apparent generalized Lamé-type equation (1.19)p with parameter $T=T(P)$.*

Recall that $P^{*}\in\Gamma(\tau,p)$ denotes the dual point of $P$. Since $T(P)=T(P^{*})$, the Baker-Akhiezer function $\psi(P^{*};z,z_{0})$ also solves the same equation (1.19)p with parameter $T(P)=T(P^{*})$. Hence $P$ and $P^{*}$ represent the same equation (1.19)p.

In the following, we collect some properties of $\psi(P;z,z_{0})$ and $\psi(P^{\ast};z,z_{0})$ associated with the equation (1.19)p. The proofs can be found in [41]. Firstly,

[TABLE]

Moreover, their Wronskian is given by

[TABLE]

where the Wronskian is defined as $W(f,g):=f^{\prime}(z)g(z)-f(z)g^{\prime}(z)$.

Different choices of $z_{0}\notin\{0,\pm p\}$ change $\psi(P;z,z_{0})$ and $\psi(P^{*};z,z_{0})$ only by nonzero multiples. We therefore omit the notation $z_{0}$ and simply write

[TABLE]

Let $P=(T,C)\in\Gamma(\tau,p)$. For $j=1,2$, set

[TABLE]

where the integration path is the fundamental cycle from $z$ to $z+\omega_{j}$ avoiding poles and zeros of $\Phi_{e}(z;T)$. Since $\Phi_{e}(z;T)$ is elliptic, the quantities $\lambda_{j}(P,z)$ are independent of $z$, and we may write:

[TABLE]

From the definition of Baker-Akhiezer functions, it follows that

[TABLE]

Thus $\psi(P;z)$ and $\psi(P^{*};z)$ are elliptic functions of the second kind. Moreover, by (4.48),

[TABLE]

The main result of this section is the following.

Theorem 4.4**.**

The generalized Lamé-type equation in the punctured non-even symmetry case (1.19)p is completely reducible if and only if $Q(T)\neq 0$.

Proof.

Let $P=(T,C)\in\Gamma(\tau,p)$. By (4.49), the Baker-Akhiezer functions $\psi(P;z)$ and $\psi(P^{*};z)$ are linearly independent if and only if $Q(T)\neq 0$. Assume first that (1.19)p is completely reducible. If $Q(T)=0$, then $\psi(P;z)$ and $\psi(P^{*};z)$ are linearly dependent, i.e.

[TABLE]

Complete reducibility guarantees the existence of two linearly independent solutions $y_{1}(z;T)$, $y_{2}(z;T)$ such that

[TABLE]

Then $y_{1}(z;T)\,y_{2}(z;T)$ is an elliptic solution of the equation (4.14). By Theorem 4.2 and (4.48),

[TABLE]

If $a$ is a zero of $\psi(P;z)$, then from (4.54) we may assume $y_{2}(a;T)=0$. Thus $y_{2}(z)=\psi(P;z)$ up to a nonzero multiple, and together with (4.54) this also implies $y_{1}(z)=\psi(P;z)$, a contradiction. Hence $Q(T)\neq 0$.

Conversely, if $Q(T)\neq 0$, then $\psi(P;z)$ and $\psi(P^{*};z)$ are linearly independent. Together with (4.51) and (4.52), this shows that (1.19)p is completely reducible. ∎

To conclude this section, we present the corresponding results for the generalized Lamé-type equation in the even-symmetric setting:

[TABLE]

where $q(z;A)$ is given in (4.8) and $B$ is determined by the apparent condition (4.9).

These results are summarized in the following theorem.

Theorem 4.5**.**

(i) Up to a nonzero multiple, there exists a unique non-trivial elliptic solution $\Phi_{e}(z;A)$ of the second symmetric product equation with respect to $q(z;A)$. The elliptic solution $\Phi_{e}(z;A)$ is precisely given by

[TABLE]

(ii) The spectral polynomial $Q(A)$ is expressed as

[TABLE]

where

[TABLE]

*(iii) The generalized Lamé-type equation in the even-symmetry case (4.55)p is completely reducible if and only if $Q(A)\neq 0$. *

5. Monodromy Theory and The Proofs

In this section we analyze the monodromy representations of of (1.19)p and present a complete proof of Theorem 1.1.

5.1. Completely Reducible Case

We establish the relation between complete reducibility and the monodromy data for the generalized Lamé-type equation (1.19)p.

Let $P\in\Gamma(\tau,p)$ and recall $\lambda_{1}(P),\,\lambda_{2}(P)$ from (4.50). Define $(r(P),s(P))\in\mathbb{C}^{2}/\mathbb{Z}^{2}$ associated with (1.19)p by

[TABLE]

It follows from (4.52) and (5.1) that

[TABLE]

Complete reducibility is characterized by the pair $(r(P),s(P))$ as follows.

Theorem 5.1**.**

Let $P=(T,C)\in\Gamma(\tau,p)$. Then $Q(T)\neq 0$ if and only if $\left(r(P),s(P)\right)\notin\frac{1}{2}\mathbb{Z}^{2}$. Namely, the generalized Lamé-type equation (1.19)p is completely reducible if and only if $\left(r(P),s(P)\right)\notin\frac{1}{2}\mathbb{Z}^{2}$.

We remark that Theorem 5.1 also holds for the generalized Lamé-type equation in the even-symmetry case (4.55)p. Here, we provide the proof for the punctured non-even symmetry case.

Proof of Theorem 5.1.

Suppose that $Q(T)\neq 0$. Then the Baker-Akhiezer functions $\psi(P;z)$ and $\psi(P^{*};z)$ are linearly independent. If $(r(P),s(P))\in\tfrac{1}{2}\mathbb{Z}^{2}$,

[TABLE]

In this case, the function

[TABLE]

is also an elliptic solution of (4.14). By Theorem 4.2, it follows that

[TABLE]

up to a nonzero multiple. Hence every zero of $\psi(P;z)$ is also a zero of $\psi(P^{*};z)$, which contradicts to the linear independence of $\psi(P;z)$ and $\psi(P^{*};z)$. Therefore, $(r(P),s(P))\notin\tfrac{1}{2}\mathbb{Z}^{2}$.

Conversely, assume $(r(P),s(P))\notin\tfrac{1}{2}\mathbb{Z}^{2}$. Without loss of generality, we may assume that $\lambda_{1}(P)\neq\pm 1$. Then

[TABLE]

which implies that $\psi(P;z)$ and $\psi(P^{*};z)$ are linearly independent by (4.52). Hence $Q(T)\neq 0$. ∎

The monodromy problem for the classical Lamé equation (1.18) has been analyzed in [44, 45], as follows.

Theorem I.* ([44, 45])* Given $(r,s)\in\mathbb{C}^{2}\setminus\mathbb{Z}^{2}$. There exists $\tilde{P}\in\tilde{\Gamma}(\tau)$ such that the classical Lamé equation (1.18) has $(r(\tilde{P}),s(\tilde{P}))=(r,s)$ if and only if there exists $\tilde{B}\in\mathbb{C}$ such that

[TABLE]

where $Q(\tilde{B})$ and $\tilde{\Gamma}(\tau)$, defined in (4.42) and (4.43), denote respectively the spectral polynomial and spectral curve of (1.18). Moreover, the solvability of the system (5.3) is further reduced to the third equation $Z(r,s,\tau)=0$.

For the last statement, suppose $Z(r,s,\tau)=0$. Let $\tilde{B}:=\wp(r+s\tau)$. Then the second equation of (5.3) follows from the classical differential equation

[TABLE]

The following result establishes the monodromy problem of the generalized Lamé-type equation in the punctured non-even symmetry case (1.19)p.

Theorem 5.2**.**

Given $(r,s)\in\mathbb{C}^{2}\setminus\mathbb{Z}^{2}$. There exists $P\in\Gamma(\tau)$ such that the generalized Lamé-type equation (1.19)p has $(r(P),s(P))=(r,s)$ if and only if there exists $T\in\mathbb{C}$ such that

[TABLE]

In view of (5.3) and (5.5), there exists a two-to-one correspondence between the parameters of the two systems:

[TABLE]

This transformation establishes a connection between the generalized Lamé-type equation (1.19)p and the classical Lamé equation (1.18). Applying this correspondence together with Theorem I and Theorem 5.2, we obtain the following result.

Theorem 5.3**.**

Given $(r,s)\in\mathbb{C}^{2}\setminus\frac{1}{2}\mathbb{Z}^{2}$. Then there exists $P=(T(P),C(P))\in\Gamma(\tau,p)$ such that the generalized Lamé-type equation (1.19)p is completely reducible with monodromy data $(r(P),s(P))=(r,s)$, if and only if there exists $\tilde{P}=(\tilde{B}(\tilde{P}),\tilde{C}(\tilde{P}))\in\tilde{\Gamma}(\tau)$ such that the classical Lamé equation (1.18) is completely reducible with monodromy data $(r(\tilde{P}),s(\tilde{P}))=(r,s)$. Moreover, the parameters $T(P)$ for (1.19)p and $\tilde{B}(\tilde{P})$ for (1.18) are related via the correspondence:

[TABLE]

Here $r(\tilde{P}),s(\tilde{P})$ is determined by the monodromy of the Baker–Akhiezer functions associated with the classical Lamé equation (1.18) at $\tilde{P}$.

To prove Theorem 5.2, we analyze the monodromy representation via the Baker-Akhiezer functions. Let $P=(T,C)\in\Gamma(\tau,p)$. From (4.51) and (5.1), we obtain

[TABLE]

Since $\psi(P;z)$ is an elliptic function of the second kind, it can be expressed as follows:

[TABLE]

where $c(P)\in\mathbb{C}$ and $a_{1}(P),a_{2}(P)\in E_{\tau}\setminus\{0\}$ are the zeros of $\psi(P;z)$.

The following proposition shows that $(r(P),s(P))$ can be determined by $a_{1}(P),a_{2}(P)$, and $c(P)$, and vice versa.

Proposition 5.4**.**

From (5.7) and (5.8), it follows that

[TABLE]

Proof.

Recall the transformation law of the Weierstrass $\sigma$function:

[TABLE]

From (5.8), we obtain

[TABLE]

Comparing with (5.7), we deduce

[TABLE]

Finally, recalling the Legendre relation

[TABLE]

we obtain (5.9). ∎

Next, we derive the algebraic equations for the zeros $a_{1}(P)$ and $a_{2}(P)$. Define

[TABLE]

where $c\in\mathbb{C}$, $a_{1},a_{2}\in E_{\tau}\setminus\{0\}$.

Theorem 5.5**.**

Let $y_{a}(z)$ be defined in (5.13).

1. [(i)]
2. (1)

Suppose $a_{1}\neq a_{2}\in E_{\tau}\setminus\{0,\pm p\}$. Then the function $y_{a}(z)$ is a solution to (1.19)p for some $T\in\mathbb{C}$, with $B$ given by (1.20), if and only if

[TABLE]

In this case, the constants $c$ and $T$ are determined by

[TABLE] 3. (2)

Suppose $a_{1}=a_{2}=\pm p$. Then $y_{a}(z)$ is a solution to (1.19)p for some $T\in\mathbb{C}$, with $B$ given by (1.20), if and only if

[TABLE]

Proof.

From the definition of $y_{a}(z)$ in (5.13), we obtain

[TABLE]

Define the elliptic function

[TABLE]

Then $y_{a}(z)$ solves (1.19)p if and only if $g(z)\equiv 0$.

Case (i). $a_{1}\neq a_{2}\in E_{\tau}\setminus\{0,\pm p\}$.

Expanding $g(z)$ at $z=0$ yields

[TABLE]

For $g$ to be holomorphic with $g(0)=0$, we must have

[TABLE]

Under (5.20), the expansions of $g$ at $z=a_{1},a_{2}$ are

[TABLE]

Thus, holomorphicity at $a_{1},a_{2}$ requires

[TABLE]

With (5.20) and (5.22), we see that

[TABLE]

near $z=p$. By (5.22), $g$ is also holomorphic at $z=p$. Consequently, $y_{a}(z)$ solves (1.19)p if and only if (5.20) and (5.22) hold, with $c$ determined by

[TABLE]

which establishes (5.14), (5.15), and (5.16).

Case (ii). $a_{1}=a_{2}\in\{\pm p\}$. The case $a_{1}=a_{2}=-p$ is obtained by replacing $p$ with $-p$. Here, we provide the proof for $a_{1}=a_{2}=p$.

Expanding $g(z)$ at $z=0$ gives

[TABLE]

Thus holomorphicity requires

[TABLE]

In this case, $g(0)=0$ holds automatically. Expanding further at $z=p$ yields

[TABLE]

forcing

[TABLE]

Hence $y_{a}(z)$ is a solution precisely when (5.24) and (5.26) hold, and

[TABLE]

verifying (5.17) and (5.18). ∎

For $P\in\Gamma(\tau,p)$, define

[TABLE]

From (5.9), we obtain

[TABLE]

Since $c(P)=\zeta(\alpha(P))\neq\infty$, $\alpha(P)=a_{1}(P)+a_{2}(P)\neq 0$. Consequently,

[TABLE]

With the above notations, we obtain the following lemma.

Lemma 5.6**.**

Let $P=(T,C)\in\Gamma(\tau,p)$. Then

[TABLE]

Proof.

We prove (5.29) and (5.30). First, assume $a_{1}(P)\neq a_{2}(P)$. Namely, $a_{1}(P),a_{2}(P)\notin\{0,\pm p\}$. By the addition formula for the Weierstrass $\wp$-function (3.27), (5.29) follows from (5.14) and (5.16). If instead $a_{1}(P)=a_{2}(P)\in\{\pm p\}$, then another addition formula gives

[TABLE]

In such case, (5.29) follows from (5.18). Finally, using the classical differential equation (5.4) together with (4.40), we obtain (5.30). ∎

Given $(r,s)\in\mathbb{C}^{2}$, set $\alpha:=r+s\tau\neq 0$. Motivated by Lemma 5.6, we next study the solvability of the system

[TABLE]

Lemma 5.7**.**

Let $(r,s)\in\mathbb{C}^{2}$ with $\alpha=r+s\tau\neq 0$.

• (i)

If $P=(T,C)\in\Gamma(\tau,p)$, then $T$ solves (5.32) with $(r,s)=(r(P),s(P))$.

• (ii)

Suppose $T$ solves (5.32). Then there exists $P=(T,C)\in\Gamma(\tau,p)$ such that $(r,s)=(r(P),s(P))$.

Proof.

Part $(i)$ follows directly from Lemma 5.6.

For part $(ii)$, let $(r,s)\in\mathbb{C}^{2}$ and suppose $T\in\mathbb{C}$ satisfies (5.32). Set $P=(T,C)\in\Gamma(\tau,p)$. Since $T$ solves (5.32), we have

[TABLE]

These equations imply

[TABLE]

Then $(r(P),s(P))=(r,s)$ follows from the Legendre relation (5.12). ∎

Proof of Theorem 5.2.

It follows directly from Lemma 5.6 and Lemma 5.7. ∎

By the correspondence (5.6), the systems (5.3) and (5.5) are equivalent. The following corollary follows from the last statement of Theorem I.

Corollary 5.8**.**

For any $(r,s)\in\mathbb{C}^{2}\setminus\tfrac{1}{2}\mathbb{Z}^{2}$, the generalized Lamé-type equation (1.19)p is completely reducible with monodromy data $(r,s)$ if and only if $Z(r,s,\tau)=0$.

5.2. Non-completely Reducible Case

To prove Theorem 1.1, it remains to treat the non-completely reducible case of equation (1.19)p.

The proofs of the following Lemmas 5.9 and 5.10 for determining the monodromy data $\mathcal{D}$ in the non-completely reducible case are essentially the same for the classical Lamé equation (1.18) and for the generalized Lamé-type equation (1.19)p. We therefore present the proofs only for (1.19)p.

Fix a base point $x_{0}\in E_{\tau}\setminus\{0,\pm p\}$, for any $T\in\mathbb{C}$ and $z$ sufficiently close to $x_{0}$, define

[TABLE]

These integrals are locally holomorphic. Since $\Phi_{e}(z;T)$ is elliptic, $\chi_{j}(z;T)$ is independent of $z$, for $j=1,2$. We denote them by $\chi_{j}(T)$.

Lemma 5.9**.**

Suppose the generalized Lamé-type equation (1.19)${}_{p,\,T_{0}}$ is non-completely reducible at parameter $T_{0}$ with monodromy data $\mathcal{D}\in\mathbb{C}\cup\{\infty\}$. Then

[TABLE]

Proof.

Since (1.19)${}_{p,\,T_{0}}$ is non-completely reducible, Theorem 5.1 implies that the Baker-Akhiezer function $\psi(P_{0};z)$ satisfies

[TABLE]

Hence $\psi(P_{0};z)^{2}$ is elliptic and solves (4.14). Consequently, up to a nonzero multiple,

[TABLE]

Let $\tilde{y}(z)$ be another solution of (1.19)${}_{p,\,T_{0}}$ linearly independent of $\psi(P_{0};z)$, which is not elliptic of the second kind. Define

[TABLE]

Substituting $\tilde{y}(z)=\chi(z)\,\psi(P_{0};z)$ into (1.19)${}_{p,\,T_{0}}$ yields

[TABLE]

Integrating (5.38) and using (5.36), we obtain

[TABLE]

for some $\tilde{b},b\in\mathbb{C}\setminus\{0\}$. Thus $\chi(z)$ is quasi-periodic. By (5.39) and (5.33), we have

[TABLE]

Note that $\chi_{1}(T_{0})$ and $\chi_{2}(T_{0})$ cannot vanish simultaneously; otherwise $\tilde{y}(z)$ would be elliptic of the second kind.

Case 1. If $\chi_{1}(T_{0})=0$, then $\chi_{2}(T_{0})\neq 0$. Therefore $\frac{\chi_{2}(T_{0})}{\,\chi_{1}(T_{0})\,}=\infty$. In this case, the fundamental solution

[TABLE]

has monodromy matrices

[TABLE]

where $\epsilon_{1},\epsilon_{2}\in\{\pm 1\}$. It follows from (5.41) that

[TABLE]

Case 2. If $\chi_{1}(T_{0})\neq 0$, then $\frac{\chi_{2}(T_{0})}{\,\chi_{1}(T_{0})\,}\in\mathbb{C}$, and

[TABLE]

forms a fundamental system. Its monodromy matrices satisfy

[TABLE]

where $\epsilon_{1},\epsilon_{2}\in\{\pm 1\}$. It follows from (5.42) that

[TABLE]

This completes the proof. ∎

Lemma 5.10**.**

Suppose the generalized Lamé-type equation (1.19)${}_{p,\,T_{0}}$ is non-completely reducible at parameter $T_{0}$ with monodromy data $\mathcal{D}\in\mathbb{C}\cup\{\infty\}$. Then $\mathcal{D}$ can be determined by

[TABLE]

Proof.

By Theorem 5.1, we have $(r(P_{0}),s(P_{0}))\in\tfrac{1}{2}\mathbb{Z}^{2}$. The polynomial $Q(T)\in\mathbb{C}[T]$ is nontrivial with $Q(T_{0})=0$. Choose any sequence $P_{k}=(T_{k},C_{k})\in\Gamma(\tau,p)$ with $C_{k}^{2}=Q(T_{k})\neq 0$ such that $T_{k}\to T_{0}$. Then the Baker-Akhiezer functions $\psi(P_{k};z)$ converge uniformly to $\psi(P_{0};z)$ on compact subsets of $E_{\tau}\setminus\{0,\pm p\}$. Consequently,

[TABLE]

For each $k$, set

[TABLE]

where $P_{k}^{*}=(T_{k},-C_{k})\in\Gamma(\tau,p)$ is the dual point of $P_{k}=(T_{k},C_{k})$. Using (5.2), one obtains

[TABLE]

Moreover, by (4.48) and (4.49), we have

[TABLE]

Integrating (5.47) along the cycles $\ell_{j}$, it follows from (5.46) that

[TABLE]

for some $m_{1}(T_{k}),m_{2}(T_{k})\in\mathbb{Z}$. Since $\chi_{j}(z;T)$ is locally holomorphic, it follows that

[TABLE]

Since $Q(T_{k})\to Q(T_{0})=0$, we have $C_{k}\to 0$, and hence

[TABLE]

Combining (5.44) and (5.50) yields

[TABLE]

for $k$ sufficiently large. Finally, by Lemma 5.9 together with (5.51),

[TABLE]

This establishes (5.43) and thereby completes the proof. ∎

5.3. The Proofs

In this subsection, we present the proofs of Theorem 1.1, Theorem 1.4 and Theorem 1.5.

Proof of Theorem 1.1.

Let $\tilde{\Gamma}(\tau)$ and $\Gamma(\tau,p)$ be the spectral curves of (1.18) and (1.19)p, respectively. Recall the correspondence

[TABLE]

which are related by

[TABLE]

Suppose the generalized Lamé-type equation (1.19)p at parameter $T_{0}$ is non-completely reducible with monodromy data $\mathcal{D}\in\mathbb{C}\cup\{\infty\}$. Let $P_{0}=(T_{0},0)\in\Gamma(\tau,p)$, and let $\tilde{P}_{0}$ denote its corresponding point on $\tilde{\Gamma}(\tau)$ via (5.52). Namely,

[TABLE]

Choose a sequence $P_{k}=(T(P_{k}),C(P_{k}))\in\Gamma(\tau,p)$ with $C(P_{k})\neq 0$ and $P_{k}\to P_{0}$ as $k\to+\infty$. Let $\tilde{P}_{k}=(\tilde{B}(P_{k}),\tilde{C}(P_{k}))\in\tilde{\Gamma}(\tau)$ denote the corresponding points of $P_{k}\in\Gamma(\tau,p)$ under (5.52). Then $\tilde{C}(P_{k})\neq 0$ and $\tilde{P}_{k}\to\tilde{P}_{0}$. Moreover,

[TABLE]

By Lemma 5.10, the monodromy data $\mathcal{D}$ is then given by

[TABLE]

The converse follows by the same argument. Therefore, in the non-completely reducible case, the classical Lamé equation (1.18) and the generalized Lamé-type equation (1.19)p are monodromy equivalent. This completes the proof. ∎

Proof of Theorem 1.4.

As in Corollary 1.3, let the parameters $T(p),B(p)$ be given by (1.21) and (1.20), respectively, so that

[TABLE]

Relation (5.54) implies

[TABLE]

We now show that $T(p)=0$ if and only if $p=\pm p_{r,s}^{(1)}(\tau)$.

Assume $T(p)=0$. Then (1.19)p reduces to the even-symmetric case. Since the monodromy data are $(r,s)$, Theorem B yields $p=p_{r,s}^{(1)}(\tau)$. Conversely, suppose $p=\pm p_{r,s}^{(1)}(\tau)$. Then $p$ satisfies (3.12). Since $(r,s)$ is the monodromy data of (1.19)p, Theorem 5.2 implies that

[TABLE]

Combining these with (3.12), we obtain

[TABLE]

and hence $T(p)=0$. This completes the proof. ∎

We investigate the limiting behavior as $p\to\tfrac{\omega_{k}}{2}$ for $k=0,1,2,3$.

Lemma 5.11**.**

Let $(T(p),B(p))$ be as in Corollary 1.3. Namely,

[TABLE]

with $B(p)$ determined by (1.20).

1. (i)

As $p\to 0$, $q(z;T(p))$ converges uniformly to $2\wp(z)+\tilde{B}$ on every compact subset of $E_{\tau}\setminus\{0\}$. 2. (ii)

For each $k=1,2,3$, as $p\to\frac{\omega_{k}}{2}$, $q(z;T(p))$ converges uniformly on every compact subset of $E_{\tau}\setminus\{0,\tfrac{\omega_{k}}{2}\}$ to

[TABLE]

where

[TABLE]

Proof.

Since $\tilde{B}$ is independent of $p$, (5.55) implies

[TABLE]

Hence, as $p\to 0$,

[TABLE]

Moreover, as $p\to 0$,

[TABLE]

and therefore

[TABLE]

It follows from (5.58)-(5.60) that, as $p\to 0$,

[TABLE]

This proves part (i). For part (ii). Set $p_{k}:=p-\tfrac{\omega_{k}}{2}$. As $p\to\tfrac{\omega_{k}}{2}$,

[TABLE]

and consequently

[TABLE]

Using (5.62) and (5.63), as $p\to\frac{\omega_{k}}{2}$,

[TABLE]

which establishes part (ii). ∎

Proof of Theorem 1.5.

Let $(T(p),B(p))$ be as in Corollary 1.3. Let $\psi_{p}(z)$ denote the the Baker-Akhiezer function of the generalized Lamé-type equation (1.19)p with parameters $(T(p),B(p))$. Since $\psi_{p}(z)$ is elliptic of the second kind, by (5.8) we have

[TABLE]

From (5.9) we have, for all $p\in E_{\tau}\setminus\{\tfrac{\omega_{k}}{2}:k=0,1,2,3\}$,

[TABLE]

Passing to a subsequence if necessary, we may assume $a_{j}^{(p)}\to a_{j}$ as $p\to\tfrac{\omega_{k}}{2}$, $j=1,2$. Let $p\to\frac{\omega_{k}}{2}$, we have

[TABLE]

uniformly on compact subsets of $E_{\tau}\setminus\{0,\frac{\omega_{k}}{2}\}$, as $p\to\tfrac{\omega_{k}}{2}$, where

[TABLE]

As $p\to\frac{\omega_{k}}{2}$, it follows from (5.66) and (5.68) that

[TABLE]

If $k=0$, the limiting function

[TABLE]

is precisely the Baker-Akhiezer function of (1.18) used to define the monodromy data $(r(\tilde{P}),s(\tilde{P}))$ for $\tilde{P}=(\tilde{B},\tilde{C})\in\tilde{\Gamma}(\tau)$, hence

[TABLE]

For $k\in\{1,2,3\}$, the Baker-Akhiezer function for the limiting equation (1.22)${}_{\tilde{B}_{k}}$, used to define the monodromy data $(r(\tilde{B}_{k}),s(\tilde{B}_{k}))$, is given by

[TABLE]

which satisfies

[TABLE]

Comparing $\psi_{k}(z)$ with $\hat{\psi}_{k}(z)$ and using the transformation law of the Weierstrass $\sigma$-function (5.10), we obtain

[TABLE]

Together with (5.69), this yields

[TABLE]

and hence

[TABLE]

∎