Algebraic geometrical solutions for certain evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians

Algebraic geometrical solutions of a new shallow-water equation and Dym-typ e equation are studied in connection with Hamiltonian flows on nonlinear su bvarieties of hyperelliptic Jacobians. These equations belong to a class of N-component integrable systems generated by Lax equations with energy-depe ndent Schrodinger operators having poles in the spectral parameter. The cla sses of quasi-periodic and soliton-type solutions of these equations are de scribed in terms of theta- and tau-functions by using new parametrizations. A qualitative description of real-valued solutions is provided.