Multidimensional dynamical systems accepting the normal shift

Dynamical systems $\ddot {\mathbf r} = \mathbf F(\mathbf r, \dot {\mathbf r})$ in $\mathbb R^n$ accepting the normal shift of submanifolds of codimension 1 are considered. The concept of weak normality for them is introduced and the defining partial differential equations for the force field $\mathbf F(\mathbf r,\dot {\mathbf r})$ of the dynamical systems with the weak and complete normality are found.