Metrisable oscillators and (super)integrable two-dimensional metrics

TL;DR

This paper introduces new classes of two-dimensional metrics that are integrable and metrisable, providing explicit geodesic expressions and connecting projective structures with integrability concepts.

Contribution

It constructs novel integrable and metrisable oscillators, introduces generalized Darboux integrability, and links projective vector fields with invariants, expanding understanding of 2D metrics.

Findings

Explicit geodesic expressions in superintegrable cases

Two families of metrics with transcendental first integrals

Identification of new integrable cases via projective Lie algebra dimensions

Abstract

We consider a family of nonlinear oscillators, which is the autonomous case of the two-dimensional projective connection. We construct several classes of these oscillators that are simultaneously integrable and metrisable. This leads to families of (super)integrable two-dimensional metrics that are parametrized by arbitrary functions. In the superintegrable case we obtain an explicit expression for the unparametrized geodesics. In the integrable case we present two families of metrics with transcendental first integrals. We introduce the concept of generalized Darboux integrability in the context of both projective equations and geodesic flows. We demonstrate that the constructed integrable metrics are generalized Darboux integrable. In addition, we establish a direct connection between relative Killing vectors and invariants of the projective vector fields that are linear in the first derivative. Finally, we compute the dimensions of the projective Lie algebra for the obtained metrics, which allows us to distinguish previously known integrable cases from new ones.