The search for critical points of fitted potential energy surfaces is facilitated by the analytical character of the function and their derivatives, but can become complicated owing to the possible presence of spurious structures generated by the fitting procedure. In this paper, we examine the capability of the gradient extremal path for following techniques in this context. In all the test cases, with this method it was possible to localize all the extrema, allowing a subsequent study of the small minima and saddle points oriented towards elucidating their spurious or true character.