Laplace Sequences of Surfaces in Projective Space and Two-Dimensional Toda Equations

We find that the Laplace sequences of surfaces of period n in projective space P_n–1 have two types, while type II occurs only for even n. The integrability condition of the fundamental equations of these two types have the same form When all _i = 1, the above equations become two-dimensional Toda equations. Darboux transformations are used to obtain explicit solutions to the above equations and the Laplace sequences of surfaces. Two examples in P₃ of types I and II are constructed.