# Symmetries and hamiltonians of Ince's XXXVIII and XLIX equations

## TL;DR

This paper analyzes the symmetries of Hamiltonians associated with Ince's I_{38} and I_{49} equations, revealing differences from Painlevé equations and enhancing understanding of their structural properties.

## Contribution

It provides a detailed study of the symmetries of Hamiltonians for Ince's I_{38} and I_{49} equations, connecting them to Weyl symmetries of Painlevé equations and highlighting key differences.

## Key findings

- Identified specific symmetry structures of I_{38} and I_{49} Hamiltonians
- Compared symmetries of Ince's equations with Painlevé equations
- Provided insights into the structural differences between these classes

## Abstract

We discuss symmetries of Hamiltonians of I$_{38}$ and I$_{49}$ equations that appear on Ince's list of fifty second-order differential equations with Painlev\'e property. This study is informed by structure of Weyl symmetries of Painlev\'e P$_{III}$ and mixed Painlev\'e P$_{III-V}$ equations and provides insights into differences between the symmetries of Painlev\'e equations and symmetries of solvable equations on Ince's list.