# On special limits of the Mixed Painlev\'e P$_{\mathbf{III-V}}$ Model

## TL;DR

This paper explores special parameter limits of the P_{III-V} Painlevé equation, revealing reductions to classical Painlevé and Ince equations, and discusses the underlying symmetry structures that differentiate these solvable cases.

## Contribution

It identifies specific parameter conditions under which P_{III-V} reduces to known equations and analyzes the symmetry groups involved, highlighting a fundamental distinction in algebraic structure.

## Key findings

- Reductions of P_{III-V} to P_{III} and Ince equations identified.
- Symmetry groups change when affine reflections are absent.
- Proposes a fundamental algebraic difference between Painlevé and other solvable equations.

## Abstract

The paper discusses P$_{III-V}$ equation for special values of its parameters for which this equation reduces to P$_{III}$, I$_{12}$, as well as, to some special cases of I$_{38}$ and I$_{49}$ equations from the Ince's list of $50$ second order differential equations possessing Painlev\'e property. These reductions also yield symmetries governing the reduced models obtained from the P$_{III-V}$ equation. We point out that the solvable equations on Ince's list emerge in this reduction scheme when the underlying reflections of the Weyl symmetry group no longer include an affine reflection through the hyperplane orthogonal to the highest root and therefore do not give rise to an affine Weyl group. We hypothesize that on the level of the underlying algebra and geometry this might be a fundamental feature that distinguishes the six Painlev\'e equations from the remaining $44$ solvable equations on the Ince's list.