A generalized TD hierarchy through the Lax matrix and their explicit solutions

Based on the resulting Lax matrix, a new finite-dimensional Hamiltonian system and a new generalized TD hierarchy of nonlinear evolution equations associated with a spectral problem are proposed. The generating function approach is used to prove the involutivity and the functional independence of conserved integrals for the finite-dimensional Hamiltonian system. The Abel-Jacobi coordinates are introduced suitably to straighten out the corresponding flows, from which algebro-geometric solutions of the generalized TD equations are derived with the help of the Riemann theta functions