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We study surfaces with parallel normalized mean curvature vector field inEuclidean or Minkowski 4-space. On any such surface we introduce specialisothermal parameters (canonical parameters) and describe these surfaces interms of three invariant functions.  We prove that any surface with parallel normalized mean curvature vectorfield parametrized by canonical parameters is determined uniquely up to amotion in Euclidean (or Minkowski) space by the three invariant functionssatisfying a system of three partial differential equations. We find examplesof surfaces with parallel normalized mean curvature vector field and solutionsto the corresponding systems of PDEs in Euclidean or Minkowski space in theclass of the meridian surfaces.

Surfaces with Parallel Normalized Mean Curvature Vector Field in Euclidean or Minkowski 4-Space