  We study the symplectic resolution of the Fano variety of lines on some
singular cyclic cubic fourfolds, i.e. cubic fourfolds arising as cyclic 3:1
cover of $\mathbb{P}^4$ branched along a cubic threefold. In particular we are
interested in the geometry of these varieties in the case of cyclic cubic
fourfolds branched along a cubic threefold having one isolated singularity of
type $A_i$ for $i=2,3,4$. On these symplectic resolutions we find a
non-symplectic automorphism of order three induced by the covering
automorphism.
