Nonexistence of an integral of the 6th degree in momenta for the Zipoy-Voorhees metric

TL;DR

This paper proves that there are no nontrivial polynomial integrals of degree less than 7 in momenta for the specific Zipoy-Voorhees spacetime with delta=2, highlighting limitations in conserved quantities.

Contribution

It establishes the nonexistence of low-degree polynomial integrals in the Zipoy-Voorhees metric, advancing understanding of integrability in this spacetime.

Findings

No polynomial integral of degree less than 7 exists for the given spacetime.

The result constrains possible conserved quantities in this gravitational field.

Supports the idea that certain spacetimes lack simple integrals of motion.

Abstract

We prove nonexistence of a nontrivial integral that is polynomial in momenta of degree less than 7 for the Zipoy-Voorhees spacetime with the parameter $\delta=2$