Minimal $P$-symmetric period problem of first-order autonomous Hamiltonian Systems

Let $P\in Sp(2n)$ satisfying $P^{k}=I_{2n}$, we consider the minimal $P$-symmetric period problem of the autonomous nonlinear Hamiltonian system \begin{equation*} \dot x(t) = JH^{\prime}(x(t)). \end{equation*} For some symplectic matrices $P$, we show that for any $\tau>0$ the above Hamiltonian system possesses a $k\tau$ periodic solution $x$ with $k\tau$ being its minimal $P$-symmetric period provided $H$ satisfies the Rabinowitz's conditions on the minimal period conjecture, together with that $H$ is convex and $H(Px)=H(x)$.