Center-Type Dynamics and Nonintegrability in Rocard’s Oscillator

Rocard’s oscillator is a third-order nonlinear equation given by z+ε ω z̈+ω ²ż +ω ³ [ε +η (1-z²-ż²ω ² ) ]z=0, z ⃛ + ε ω z ¨ + ω 2 z ˙ + ω 3 [ ε + η ( 1 - z 2 - z ˙ 2 ω 2 ) ] z = 0 , depending on the parameters ε ε , η η and ω ≠ 0 ω ≠ 0 , where ω ω sets a characteristic time scale and is treated as fixed throughout the analysis. We prove that, for η = 0 η = 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 ε = 0 ε = 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 η =0 η = 0 with the conservative reversible structure arising in the dissipation-free regime.