Generalized Weierstrass representation of surfaces in R4

In the present paper, we describe the conformal immersion of the surface into R4 by means of a linear system. Furthermore we prove that every regular conformal immersion of a surface into R4 is locally determined by the generalized Weierstrass formulae. We also give the representation of the surface with parallel mean curvature vector by solutions with the parameter of a linear system which is determined by the sinh–Laplace equation.