On the eigenpoints of cubic surfaces

We show that the eigenschemes of $4\times 4\times 4$ symmetric tensors are parametrized by a linear subvariety of the Grassmannian $\mathrm{Gr}(3,{\mathbb{P}}^{14})$. We also study the decomposition of the eigenscheme into the subscheme associated to the zero eigenvalue and its residue. In particular, we categorize the possible degrees and dimensions.