The Calogero equation and Liouville type equations

TL;DR

This paper introduces a two-component generalization of the Calogero equation, demonstrating its integrability and solvability via reciprocal transformations, and derives a generalized Liouville equation dependent on arbitrary functions.

Contribution

It presents a novel two-component integrable extension of the Calogero equation and links it to a generalized Liouville equation through reciprocal transformations.

Findings

The two-component Calogero system is C-integrable.

Both equations are solvable via reciprocal transformations to ODEs.

A generalized Liouville equation with arbitrary functions is derived.

Abstract

In this paper we present a two-component generalization of the C-integrable Calogero equation (see [1]). This system is C-integrable as well, and moreover we show that the Calogero equation and its two-component generalization are solvable by a reciprocal transformation to ODE's. Simultaneously we obtain a generalized Liouville equation (34), determined by two arbitrary functions of one variable.