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 partial derivative (2)omega (i)/partial derivativex partial derivativet = -alpha (i-1)e(omegai-1) + 2 alpha (i)e(omegai) - alpha (i+1)e(omegai+1), alpha (i) = +/-1 (i = 1, 2, ..., n). When all alpha (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 P-3 of types I and II are constructed.