On integral conditions for the existence of first integrals analytic saddle singularities

TL;DR

This paper establishes integral conditions under which analytic saddle singularities admit holomorphic first integrals, linking their existence to the vanishing of specific line integrals on fibers, with applications to quadratic singularities.

Contribution

It provides a sharp criterion connecting the existence of first integrals to line integral vanishing conditions for deformations of saddle singularities.

Findings

Existence of first integrals is equivalent to vanishing line integrals on fibers.

Results are valid under nondegeneracy conditions of the singular set.

Criteria are applicable to quadratic analytic center-cylinder singularities.

Abstract

We study one-parameter analytic integrable deformations of the germ of $2\times(n-2)$-type complex saddle singularity given by $d(xy)=0$ at the origin $0 \in \mathbb C^2\times \mathbb C^{n-2}$. Such a deformation writes ${\omega}^t=d(xy) + \sum\limits_{j=1}^\infty t^j \omega_j$ where $t\in \mathbb C,0$ is the parameter of the deformation and the coefficients $\omega_j$ are holomorphic one-forms in some neighborhood of the origin $0\in \mathbb C^n$. We prove that, under a nondegeneracy condition of the singular set of the deformation, with respect to the fibration $d(xy)=0$, the existence of a holomorphic first integral for each element ${\omega}^t$ of the deformation is equivalent to the vanishing of certain line integrals $\oint_{\gamma_c}{\omega}^t=0, \forall \gamma_c, \forall t$ calculated on cycles $\gamma_c$ contained in the fibers $xy=c, \,0 \ne c \in \mathbb C,0$. This result is quite sharp regarding the conditions of the singular set and on the vanishing of the integrals in cycles. It is also not valid for ramified saddles, i.e., for deformations of saddles of the form $x^ny^m=c$ where $n+m>2$. As an application of our techniques we obtain a criteria for the existence of first integrals for integrable codimension one deformations of quadratic analytic center-cylinder type singularities in terms of the vanishing of some easy to compute line integrals.