Rocard’s oscillator is a third-order nonlinear equation given by \( \dddot{z}+\varepsilon \omega \ddot{z}+\omega ^{2}\dot{z} +\omega ^{3}\Bigl [\varepsilon +\eta \Bigl (1-z^{2}-\frac{\dot{z}^{2}}{\omega ^{2}}\Bigr )\Bigr ]z=0, \) depending on the parameters \(\varepsilon \) , \(\eta \) and \(\omega \ne 0\) , where \(\omega \) sets a characteristic time scale and is treated as fixed throughout the analysis. We prove that, for \(\eta = 0\) , the equilibrium at the origin exhibits center-type behavior on a two-dimensional center manifold and admits a local analytic first integral. This explains the vanishing of the first Lyapunov coefficient and rules out a Hopf bifurcation at this parameter value. We then analyze the conservative limit \(\varepsilon = 0\) , showing that the system is reversible and volume-preserving, in a setting compatible with the nonintegrability results of Hu and Tang (2025). Finally, we discuss these nonintegrability results within a geometric framework connecting the center-type dynamics at \(\eta =0\) with the conservative reversible structure arising in the dissipation-free regime.
