On the subset of normality equations describing generalized Legendre   transformation
  Normality equations describe Newtonian dynamical systems admitting normal shift of hypersurfaces. They were first derived in Euclidean geometry, then in Riemannian geometry. Recently they were rederived in more general case, when geometry of manifold is given by generalized Legendre transformation. As appears, in this case some part of normality equations describe generalized Legendre transformation itself irrespective to that Newtonian dynamical system, for which others are written. In present paper this smaller part of normality equations is studied. 
Ruslan Sharipov