There are such systems that can be solved in some sense although it is not expected to be solved easily, or that various properties can be “well clarified”. For example, the soliton equations describing the nonlinear waves can be solved exactly although they are nonlinear partial differential equations which are usually difficult to handle. For the Painlevé equations, a certain family of nonlinear ordinary differential equations, it is proved rigorously that they cannot be solved in general. In addition, their underlying symmetries are high and they admit a family of good solutions in certain special locations in the parameter spaces. Behind such phenomena there lie miraculous and universal mathematical machineries which are involved with wide range of mathematics in the deep background. I am studying underlying mathematical structures of such systems called the “integrable systems”, and seeking generalization and application of their machineries. In particular, my intense interest lies in discrete (difference equations) and ultradiscrete (cellular automata) integrable systems. Recently I am developing applications of those theories to “good” discretization of curves and surfaces in Euclidean space.