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In the present paper, we study geodesic mappings of special pseudo-Riemannian manifolds called <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>V</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>K</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>-spaces. We prove that the set of solutions of the system of equations of geodesic mappings on <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>V</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>K</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>-spaces forms a special Jordan algebra and the set of solutions generated by concircular fields is an ideal of this algebra. We show that pseudo-Riemannian manifolds admitting a concircular field of the basic type form the class of manifolds closed with respect to the geodesic mappings.