# Non-Integrability of Geodesic dynamics of Chazy-Curzon space-time

## TL;DR

This paper investigates the geodesic equations of Chazy-Curzon space-time and demonstrates their non-integrability by analyzing the absence of additional analytic first integrals around specific equilibrium points.

## Contribution

It provides a novel analysis showing that the geodesic system in Chazy-Curzon space-time lacks extra integrals, indicating non-integrability.

## Key findings

- No additional analytic first integral exists for the system.
- Periodic solutions are only found at specific equilibrium points.
- The period function's properties imply non-integrability.

## Abstract

We study the integrability of the geodesic equations of the Chazy- Curzon space-time. It was established that for the equilibrium point $p_{\rho}=p_z=z=0$ and, $\rho_0 \in (1,\, 2)$, there are only periodic solutions, the Hamiltonian system, describing geodesic motion of Chazy-Curzon space-time has no additional analytic first integral. Our approach is based on the following: if the system has a family of periodic solutions around an equilibrium and if the period function is infinitely branched then the system has no additional analytical first integral.