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

We study the integrability of the geodesic equations of the Chazy- Curzon space-time. It was established that for the equilibrium point $p_ρ=p_z=z=0$ and, $ρ_0 \in (1,\, 2)$, 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.

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