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Abstract. The dynamical systems of the formr = F(r,ṙ) in 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 F(r,ṙ) of the dynamical systems with weak and complete normality are derived.

Tzitzeica’s class of surfaces in R3 is considered in connection with the inverse scattering method for the associated equation uxy=eu−e−2u. The Bäcklund transformation for this equation is derived from Tzitzeica’s Darboux transformation for it.

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