  We show that Spaltenstein varieties of classical groups are pure dimensional
when the Jordan type of the nilpotent element involved is an even or odd
partition. We further show that they are Lagrangian in the partial resolutions
of the associated nilpotent Slodowy slices, from which their dimensions are
known to be one half of the dimension of the partial resolution minus the
dimension of the nilpotent orbit. The results are then extended to the
$\sigma$-quiver-variety setting.
