On the solution set of a nonlinear 2nd order ode

Given $V \in C^1(\mathbb{R}^d)$ we consider the 2nd order ODE $$\tag{1} \ddot{x}+\nabla V(x)=0. $$ Suppose $\nabla V(0)=0$ and the Hessian $A=\text{Hess}V(0)$ is positive definite. Then the solution set $S_0$ of the linearized ODE $$\tag{2} \ddot{x}+Ax=0 $$ is a vector space of dimension $k=\dim S_0 \in \{2,\ldots,2d\}$. The period of each member of the basis of $S_0$ depends on $A$.

If (1) possesses a non-empty solution set $S$ of periodic solutions, say with period $T>0$, what could said about the structure of $S$? Does it have some kind of manifold structure? (In Structure of the solution set of a 2nd order ODE I asked a different question).

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