On two-dimensional integrable models with a cubic or quartic integral of motion

TL;DR

This paper constructs new two-dimensional integrable models with cubic or quartic integrals of motion, based on a prepotential satisfying a nonlinear PDE, revealing symmetries and parameter dependencies.

Contribution

It introduces a method to generate integrable models with higher-order integrals using symmetry considerations and a single prepotential.

Findings

New models with cubic or quartic integrals constructed

Models involve one or two continuous parameters

A conjecture on hidden dihedral symmetry for higher-order integrals

Abstract

Integrable two-dimensional models which possess an integral of motion cubic or quartic in velocities are governed by a single prepotential, which obeys a nonlinear partial differential equation. Taking into account the latter's invariance under continuous rescalings and a dihedral symmetry, we construct new integrable models with a cubic or quartic integral, each of which involves either one or two continuous parameters. A reducible case related to the two-dimensional wave equation is discussed as well. We conjecture a hidden D_{2n} dihedral symmetry for models with an integral of n-th order in the velocities.