A note on the structure of prescribed gradient--like domains of non--integrable vector fields

Given a geometric structure on $\mathbb{R}^{n}$ with $n$ even (e.g. Euclidean, symplectic, Minkowski, pseudo-Euclidean), we analyze the set of points inside the domain of definition of an arbitrary given $\mathcal{C}^1$ vector field, where the value of the vector field equals the value of the left/right gradient--like vector field of some fixed $\mathcal{C}^2$ potential function, although a non-integrability condition holds at each such a point. Particular examples of gradient--like vector fields include the class of gradient vector fields with respect to Euclidean or pseudo-Euclidean inner products, and the class of Hamiltonian vector fields associated to symplectic structures on $\mathbb{R}^{n}$ (with $n$ even). The main result of this article provides a geometric version of the main result of [1].