Lie algebras of projective motions of five-dimensional $h$-spaces $H_{221}$ of type $\{221\}$

We study five-dimensional pseudo-Riemannian $h$-spaces $H_{221}$ of type $\{221\}$. Necessary and sufficient conditions are determined under which $H_{221}$ is a space of constant (zero) curvature. Non-homothetical projective motions in $H_{221}$ of non-constant curvature are found, homotheties and isometries of the indicated spaces are investigated. Dimensions, basic elements, and structure equations of maximal projective Lie algebras acting in $H_{221}$ of non-constant curvature are determined. As a result, the classification of $h$-spaces $H_{221}$ of type $\{221\}$ by (non-homothetical) Lie algebras of infinitesimal projective and affine transformations is obtained.