Quasigraded Lie algebras on hyperelliptic curves and classical integrable systems

A new family of infinite-dimensional Lie algebras on hyperelliptic curves is constructed. We show them to be quasigraded and explicitly find their central extensions. We also show, that constructed algebras in the case of zero central charge possess infinite number of invariant functions. Besides, they admit a decomposition into the direct sum of two subalgebras. These two facts together enables one to use them to construct new integrable systems. In such a way we find new integrable Hamiltonian systems, which are direct higher rank generalizations of the integrable systems of Steklov-Liapunov, associated with the e(3) algebra and Steklov-Veselov associated with the so(4) algebra. Besides we give hyperelliptic Lax representation for the generalized Euler tops. (C) 2001 American Institute of Physics.