  We study lines and twisted cubics on cubic fourfolds with simple isolated
singularities. We show that the Hilbert scheme compactification of the total
space of Starr's fibration on the space of twisted cubics on a cubic fourfold
with simple isolated singularities and not containing a plane admits a
contraction to a singular projective symplectic variety of dimension eight. The
latter admits a crepant resolution which is a deformation of the symplectic
eightfold constructed from the space of twisted cubics on a smooth cubic
fourfold. Thereby we give another proof of the fact that the latter is a
deformation of the Hilbert scheme of four points on a K3 surface. We also show
that the variety of lines on a cubic fourfold with simple singularities is a
singular symplectic variety birational to the Hilbert scheme of two points on a
K3 surface.
