  This article is motivated by the original Casson invariant regarded as an
integral lifting of the Rochlin invariant. We aim to defining an integral
lifting of the Adams e-invariant of stably framed 3-manifolds, perhaps endowed
with some additional structure. We succeed in doing so for manifolds which are
links of normal complex Gorenstein smoothable singularities. These manifolds
are naturally equipped with a canonical $\SU$-frame. To start we notice that
the set of homotopy classes of $\SU$-frames on the stable tangent bundle of
every closed oriented 3-manifold is canonically a $\mathbb Z$-torsor. Then we
define the $\widehat E$-invariant for the manifolds in question, an integer
that modulo 24 is the Adams e-invariant. The $\widehat E$-invariant for the
canonical frame equals the Milnor number plus 1, so this brings a new viewpoint
on the Milnor number of the smoothable Gorenstein surface singularities.
