Explicit solutions of the 2+1-dimensional modified Toda lattice through straightening out of the relativistic Toda flows

The 2 + 1-dimensional modified Toda lattice is decomposed into solvable ordinary differential equations with the help of the 1 + 1-dimensional relativistic Toda lattices. Based on the decomposition and the theory of algebraic curve, the straightening out of various flows, including the continuous flow and discrete flow, is exactly given through the introduced Abel-Jacobi coordinates. The explicit theta function solutions for the 2 + 1-dimensional modified Toda lattice are obtained explicitly.