Harmonic mappings of two-dimensional fields connected to 2-family of planes in E/sub 4/ invariantly

Two-dimensional manifold V_2,2 ¹ of planes L ₂ ¹ is studied in four-dimensional Euclidean space E₄. The two-dimensional manifold V_2,2 ² of planes L₂ ² associates with this manifold. The planes are, orthogonal to the appropriate planes L₂ ¹ of the points A and equipping planes of manifold V_2,2 ¹. There are mappings between the appropriate planes L₂ ¹ isin V_2,2 ¹ and L₂ ² isin V_2,2 ². The geometrical sense of these displays is found out. Particular cases are found when the given mappings satisfy to either Cauchy-Riemann or d´Alambert-Euler conditions or they are harmonious in some or all points of the corresponding planes L₂ ¹ or L₂ ² . All considerations have local character, and the mappings are assumed analytical