Dynamical systems that admit normal displacement

The classical geometrical construction of Bianchi–Lie, Bäcklund, and Darboux transformations is considered and generalized for dynamical systems. For a transformation that generalizes normal displacement, a class of dynamical systems that admit this transformation is found. A differential equation that distinguishes dynamical systems in $\mathbb R^2$ that belong to this class is derived, and some solutions of it are considered.