Equilibria of vortex type Hamiltonians on closed surfaces

We prove the existence of critical points vortex type Hamiltonians \[ H(p_1,\ldots, p_N) = \sum_{{i,j=1}\atop{i\ne j}} ^N \Gamma_i\Gamma_jG(p_i,p_j)+\Psi(p_1,\dots,p_N) \] on a closed Riemannian surface $(\Sigma,g)$ which is not homeomorphic to sphere or projective plane. Here $G$ denotes Green function Laplace-Beltrami operator in $\Sigma$, $\Psi\colon \Sigma^N\to\mathbb{R}$ may be any class ${\mathcal C}^1$, and $\Gamma_1,\dots,\Gamma_N\in\mathbb{R}\setminus\{0\}$ are vorticities. The Kirchhoff-Routh Hamiltonian from fluid dynamics corresponds $\Psi(p) -\sum\limits_{i=1}^N \Gamma_i^2h(p_i,p_i)$ where $h\colon \Sigma\times\Sigma\to\mathbb{R}$ regular part operator. obtain $p=(p_1,\dots,p_N)$ for arbitrary $N$ vorticities $(\Gamma_1,\dots,\Gamma_N)$ $\mathbb{R}^N\setminus V$ $V$ an explicitly given algebraic variety codimension 1.