Integrable geodesic flows and Multi-Centre versus Bianchi A metrics

TL;DR

The paper explores the relationship between Multi-Centre metrics with integrable geodesic flows and Bianchi type A metrics, identifying new integrable cases and analyzing coordinate separability issues.

Contribution

It demonstrates that many Multi-Centre metrics with integrable geodesic flows are Bianchi type A, introduces new non-diagonal Bianchi metrics, and clarifies coordinate separation problems.

Findings

Most Multi-Centre metrics with integrable geodesic flow are Bianchi type A.

Diagonal bi-axial Bianchi II metrics have integrable geodesic flows.

The hyperkähler Bianchi II metric exhibits a finite-dimensional W-algebra.

Abstract

It is shown that most, but not all, of the four dimensional metrics in the Multi-Centre family with integrable geodesic flow may be recognized as belonging to spatially homogeneous Bianchi type A metrics. We show that any diagonal bi-axial Bianchi II metric has an integrable geodesic flow, and that the simplest hyperk\"ahler metric in this family displays a finite dimensional W-algebra for its observables. Our analysis puts also to light non-diagonal Bianchi VI$_0$ and VII$_0$ metrics which seem to be new. We conclude by showing that the elliptic coordinates advocated in the literature do not separate the Hamilton-Jacobi equation for the tri-axial Bianchi IX metric.