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. 
A.Yu.Boldin and R.A.Sharipov (Department of Mathematics, Bashkir State   University, Frunze str. 32, 450074 Ufa, Russia)