On a problem of È. Cartan (Q1374967)

\textit{É. Cartan} [Bull Soc. Math. France 52, 205-241 (1924; JFM 50.0500.02)] introduced the notion of an \(n\)-dimensional space with a projective connection and showed that the geodesic lines of this space can be considered as integral lines of a special quasigeodesic flow (QF) on on an \((n-1)\)-dimensional manifold \(M_{n-1}\), that is, a second-order ODE \(\frac{\partial^2 x^i}{\partial t^2} = - P^i (\frac{\partial x^i}{\partial t}) +\frac{\partial x^i}{\partial t} P^n (\frac{\partial^2 x^j}{\partial t})\), where \(x^i, 1 \leq i, j \leq n - 1\), and \(P^n\) are second-degree polynomials (not necessarily homogeneous) of the ``speed'' \(\frac{\partial^2 x^j}{\partial t}\) whose coefficients are functions of \(x^j\) and \(t\). In the same paper, Cartan posed the problem to find a generalization of a space with a projective connection such that its geodesic lines can provide a model of integral curves of any QF, \(\frac{\partial^2 x^i}{\partial t^2} = D^i (t, x^j, \frac{\partial x^i}{\partial t})\) on \(M_{n-1}\). In the paper under review the author presents a solution of this problem of Cartan.