On the properties of the projective Lie algebras of rigid $h$-spaces $H_{32}$ of the type $\{32\}$

The five-dimensional pseudo-Riemannian spaces that admit infinitesimal projective transformations were studied. A general solution of the Eisenhart equation in the $h$-space $H_{32}$ of non-constant curvature was found. Necessary and sufficient conditions were established for the existence of a non-homothetic projective motion in the $h$-space $H_{32}$ of non-constant curvature, and, as a consequence, the structure of the non-homothetic projective Lie algebra in this space was determined.