Multidimensional Dynamical Systems Accepting the Normal Shift

The dynamical systems of the form \ddot\bold r=\bold F (\bold r,\dot\bold r) in $\Bbb R^n$ accepting the normal shift are considered. The concept of weak normality for them is introduced. The partial differential equations for the force field \bold F(\bold r,\dot\bold r) of the dynamical systems with weak and complete normality are derived.