Projective analog of Egorov transformation

We prove the following assertion, which is a projective analog of the well-known Egorov theorem on surfaces in the Euclidean space: a family of lines $v=\mathrm{const}$ on a surface $S$ in $\mathbf P^3$ is a basis for Egorov transformation if and only if the surface bands defined on $S$ by these lines belong to bilinear systems of plane elements. There exist a whole set of Egorov transformations that depend on one function of $v$ with this family of lines as the basis of the correspondence.