r-matrix and algebraic-geometric solution for integrable symplectic map

A new Lax matrix is introduced for the integrable symplectic map (ISM), and the non-dynamical (i.e. constant)r-matrix of ISM is obtained. Moreover, an effective approach is systematically presented to construct the explicit solution (here, the explicit solution means algebraic-geometric solution expressed by the Riemann-Theta function) of a soliton system or nonlinear evolution equation from Lax matrix,r-matrix, and the theory of nonlinearization through taking the Toda lattice as an example. The given algebraic-geometric solution of the Toda lattice is almost-periodic and includes the periodic and finite-band solution.