Abstract <p>A heuristic method that allows us to determine in advance the cases of integrability of autonomous dynamical systems with a polynomial right-hand side is used. The capabilities of this method are demonstrated using examples of two- and three-dimensional dynamical systems with a quadratic nonlinearity. A significant achievement compared to previous works is the ability to study systems of a general type without resonances in the linear parts, which is achieved by generalizing the results of resonance cases. Thus, it becomes possible to use the obtained results when working with dynamical models of real systems.</p>