Extension of constrained algebra for closed string with the world surface of fixed topology

The recently suggested gauge in which the constraints and additional conditions form a closed algebra is extended to the case of the Krichever-Novikov generally graded algebras. We show that the central element of the extended algebra is presented by a nonexact form on the closed contour of the world surface of the string. Realization of this algebra in terms of the string variables is presented. The classical dynamics of the closed boson string with the world surface of fixed genus is considered. The dynamical variables are introduced in a covariant way. The Hamilton equations are written in terms of these variables. The obtained equations are equivalent to Lagrange equations only in the case of harmonic time function