  We study the moments finiteness problem for the class of Lipschitz maps $F:
[a,b]\rightarrow\mathbb R^n$ with images in a compact Lipschitz triangulable
curve $\Gamma$. We apply the obtained results to the center problem for ODEs
describing in some cases (including equations with analytic coefficients) the
set of universal centers of such equations by vanishing of finitely many
moments from their coefficients.
