  In this article, it is proved that the eigenvalue variety of the exterior of
a nontrivial, non-Hopf, Brunnian link in $\mathbb{S}^{3}$ contains a nontrivial
component of maximal dimension. This generalises, for Brunnian links, the
nontriviality of the $A$-polynomial of a nontrivial knot in $\mathbb{S}^{3}$.
