Eigenpoint collinearities of plane cubics

Given a ternary homogeneous polynomial, the fixed points of the map from $\mathbb{P}^2$ to itself defined by its gradient are called its eigenpoints. We focus on cubic polynomials, and analyze configurations of eigenpoints that admit one or more alignments. We give a classification and explicit equations, in the coordinates of the points, of all configurations: this is accomplished by using both geometric techniques and by an extensive use of computer algebra.