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 domain definition an arbitrary given $\mathcal{C}^1$ vector field, where value field equals left/right gradient--like some fixed $\mathcal{C}^2$ potential function, although non-integrability condition holds at each such point. Particular examples fields include class gradient respect to Euclidean or pseudo-Euclidean inner products, and Hamiltonian associated symplectic structures (with even). The main result this article provides version [1].