The geodesic flow of the BGPP metric is Liouville integrable

TL;DR

This paper proves that the geodesic equations for the BGPP metric are integrable in the Liouville sense, enabling explicit solutions or reduction to quadratures, especially in degenerate cases like the Eguchi-Hanson metric.

Contribution

It demonstrates the Liouville integrability of the BGPP metric's geodesic flow and provides explicit solutions or reductions for the system.

Findings

Geodesic equations for BGPP metric are Liouville integrable.

Reduction from four to two degrees of freedom using SO(3,R) symmetry.

Explicit solutions or quadrature reductions, including elliptic integrals for degenerate cases.

Abstract

We prove that the geodesics equations corresponding to the BGPP metric are integrable in the Liouville sense. The $\mathrm{SO}(3,\mathbb{R})$ symmetry of the model allows to reduce the system from four to two degrees of freedom. Moreover, solutions of the reduced system and its degenerations can be solved explicitly or reduced to a certain quadrature. In degenerated cases BGPP metric coincides with the Eguchi-Hanson metric and for this case the mentioned quadrature can be calculated explicitly in terms of elliptic integrals.