On the local coordinates on the manifold of finite-gap solutions of the KdV equation

A theorem is proved about nondegeneracy of the map
$$ (E_1<E_2<\dots<E_{2g+1})\to(V,W,c), $$
where $E_i$ are the branching points of the hyperelliptic curve $\Gamma$, which corresponds to the finite-gap solution of KdV equation $u_g(x,t)$. Here $V,W$ are frequency vectors and $c$ is the “mean value” of the potential $u_g(x,t)$. The bijectivity of this map for $g=1$ is proved. Complex generalization of the nondegeneracy result is proved. Bibliography: 11 titles.