It is proved that every solution of the stationary Toda equation in the generic case is represented as a discrete orbit of the symplectic map obtained through nonlinearization of the Toda eigenvalue problem. As an application, the calculation of the finite-band solution of the Toda lattice equation is reduced to the solution of a system of ODES plus a simple iterative process of the symplectic map.
Ragnisco, O., Cao, C., Wu, Y.t. (1995). ON THE RELATION OF THE STATIONARY TODA EQUATION AND THE SYMPLECTIC MAPS. JOURNAL OF PHYSICS. A, MATHEMATICAL AND GENERAL, 28(3), 573-588 [10.1088/0305-4470/28/3/013].