Projective geometry of systems of second-order differential equations

It is proved that every projective connection on an $n$-dimensional manifold $M$ is locally defined by a system $\mathscr S$ of $n-1$ second-order ordinary differential equations resolved with respect to the second derivatives and with right-hand sides cubic in the first derivatives, and that every differential system $\mathscr S$ defines a projective connection on $M$. The notion of equivalent differential systems is introduced and necessary and sufficient conditions are found for a system $\mathscr S$ to be reducible by a change of variables to a system whose integral curves are straight lines. It is proved that the symmetry group of a differential system $\mathscr S$ is a group of projective transformations in $n$-dimensional space with the associated projective connection and has dimension $\leqslant n^2+2n$. Necessary and sufficient conditions are found for a system to admit the maximal symmetry group; basis vector fields and structure equations of the maximal symmetry Lie algebra are produced. As an application a classification is given of the systems $\mathscr S$ of two second-order differential equations admitting three-dimensional soluble symmetry groups.
Bibliography: 22 titles.