The Riemann extensions in theory of differential equations and their applications

Some properties of the $4$-dim Riemannian spaces with the metrics
$$ ds^2=2(za_3-ta_4)dx^2+4(za_2-ta_3)dxdy+2(za_1-ta_2)dy^2+2dxdz+2dydt $$
connected with the second order nonlinear differential equations
\begin{equation} y''+a_{1}(x,y){y'}^3+3a_{2}(x,y){y'}^2+3a_{3}(x,y)y'+a_{4}(x,y)=0 \tag{1} \end{equation}
with arbitrary coefficients $a_{i}(x,y)$ are studied. The properties of dual equations for the equations (1) are considered. The theory of the invariants of second order ODE's for investigation of the nonlinear dynamical systems with parameters is used. The property of the eight dimensional extensions of the four-dimensional Riemannian spaces of General Relativity are discussed.