N. S. Sinyukov [5] introduced the concept of an almost geodesic mapping of a space $A_n$ with an affine connection without torsion onto ${\overline{A}}_n$ and found three types: ${\pi }_1$, ${\pi }_2$ and ${\pi }_3$. The authors of [1] proved completness of that classification for $n>5$.By definition, special types of mappings ${\pi }_1$ are characterized by equations $$P_{ij,k}^h+P_{ij}^{\alpha }{P}_{\alpha k}^h=a_{ij}{\delta }_k^h,$$ where $P_{ij}^h\equiv {\overline{\Gamma }}_{ij}^h-{\Gamma }_{ij}^h$ is the deformation tensor of affine connections of the spaces $A_n$ and ${\overline{A}}_n$.In this paper geometric objects which preserve these mappings are found and also closed classes of such spaces...