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Abstract. In the work [10] we obtained derivational equations of a submanifold of a space L N with asymmetric affine connection. Based on asymmetry of the connection we define four kinds of covariant derivative and obtain four kinds of derivational equations.In [20] are examined integrability conditions of derivational equations, using the 1 st and the 2 nd kind of derivative, and in the present work we do it on the base of the 3 rd and 4 t h kind.2010 Mathematics Subject Classification: 53C25; 53A45; 53B05.

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In the present paper a generalized Kählerian space GK 1 N of the first kind is considered, as a generalized Riemannian space GR N with almost complex structure F h i , that is covariantly constant with respect to the first kind of covariant derivative.Using the non-symmetric metric tensor we find necessary and sufficient conditions for a geodesic mapping f : GR N → GK 1 N with respect to the four kinds of covariant derivatives.