Local integrals generating exact integrals of motion in three-dimensional potential fields: part I

The problem of identifying additional local integrals of motion in generic three-dimensional potentials remains a fundamental issue in dynamical systems and galactic dynamics, since most realistic gravitational fields lack global symmetries or separable coordinate systems. In this study, we propose a general constructive method for deriving such local integrals by introducing auxiliary functions whose gradients define an orthogonal frame adapted to the potential. The central idea is that the velocity field can be characterized by two quadratic algebraic relations whose compatibility conditions uniquely determine the orthogonal frame and ensure the existence of two additional isolating integrals. Our main results include: (i) the derivation of the complete set of necessary and sufficient compatibility conditions for these auxiliary functions; (ii) the proof that these functions generate two independent local quadratic integrals in addition to the energy integral; and (iii) the explicit construction of the associated orthogonal coordinate systems. The novelty of the proposed method lies in its unified nonperturbative framework, which does not rely on global separability, in contrast to classical Stäckel theory and existing approximation-based approaches. Applications to ellipsoidal and spherical coordinate systems demonstrate that the method recovers known global integrals in separable cases while also yielding new classes of local integrals for more general potentials. Consequently, the proposed framework extends the class of partially integrable models relevant to celestial mechanics and stellar dynamics.