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 Pn−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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\frac{{\partial ^2 \omega _i }}{{\partial x\partial t}} = - \alpha _{i - 1} {\text{e}}^{\omega _{{\text{i - 1}}} } + 2\alpha _i {\text{e}}^{\omega _i } - \alpha _{i + 1} {\text{e}}^{\omega _{{\text{i + 1}}} } ,{\text{ }}\alpha _i = \pm 1{\text{ }}(i = 1,2, \cdots ,n).$$ \end{document} When all αi = 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.