  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.
