On the solution of normality equations for the dimension $n\geq 3$

The normality equations for the Newtonian dynamical systems on an arbitrary Riemannian manifold of the dimension $n \geq 3$ are considered. Locally the solution of such equations reduces to three possible cases: in two of them the solution is written out explicitly, and in the third case the equations of normality are reduced to an ordinary differential equation of the second order. Some new examples of explicit solutions of normality equations are constructed.