  For modelling of various physical processes, geodesic lines and almost
geodesic curves serve as a useful tool. Trasformations or mappings between
spaces (endowed with a metric or connection) which preserve such curves play an
important role in physics, particularly in mechanics, and in geometry as well.
Our aim is to continue investigations concerning existence of almost geodesic
mappings of manifolds with linear (affine) connection, particularly of the
so-called ${\tilde \pi}_1$ mappings, i.e. canonical almost geodesic mappings of
type $\pi_1$ according to Sinyukov. First we give necessary and sufficient
conditions for existence of ${\tilde \pi}_1$ mappings of a manifold endowed
with a linear connection onto pseudo-Riemannian manifolds. The conditions take
the form of a closed system of PDE's of first order of Cauchy type. Further we
deduce necessary and sufficient conditions for existence of ${\tilde \pi}_1$
mappings onto generalized Ricci-symmetric spaces. Our results are
generalizations of some previous theorems obtained by N.S. Sinyukov.
