critical points for the energy functional

We study the equilibrium states for an energy functional with a parametric force field on a region of a surface. Consideration of free equilibrium states is based on Lyusternik - Schnirelman''s and Skrypnik''s variational methods. Consideration of equilibrium states under a constraint of geometrical character is based on an analog of Skrypnik''s method, described in cite{vyr3}. In local coordinates, equilibrium points satisfy an elliptic boundary value problem.