  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.
