Nonexistence of the final first integral in the Zipoy-Voorhees space-time

TL;DR

This paper proves that the geodesic equations in the Zipoy-Voorhees space-time lack an extra meromorphic first integral, indicating non-integrability of the system.

Contribution

It demonstrates the nonexistence of a meromorphic first integral for geodesic motion in Zipoy-Voorhees space-time, establishing its non-Liouville integrability.

Findings

Geodesic motion in Zipoy-Voorhees space-time is not Liouville integrable.

No additional meromorphic first integral exists for this space-time.

The system's dynamics are non-integrable.

Abstract

We show that the geodesic motion in the Zipoy-Voorhees space-time is not Liouville integrable, in that there does not exist an additional first integral meromorphic in the phase space variables.