  For each real number $\Lambda$ a Lie algebra of nonlinear vector fields on
three dimensional Euclidean space is reported. Although each algebra is
mathematically isomorphic to $gl(3,{\bf R})$, only the $\Lambda=0$ vector
fields correspond to the usual generators of the general linear group. The
$\Lambda < 0$ vector fields integrate to a nonstandard action of the general
linear group; the $\Lambda >0$ case integrates to a local Lie semigroup. For
each $\Lambda$, a family of surfaces is identified that is invariant with
respect to the group or semigroup action. For positive $\Lambda$ the surfaces
describe fissioning nuclei with a neck, while negative $\Lambda$ surfaces
correspond to exotic bubble nuclei. Collective models for neck and bubble
nuclei are given by irreducible unitary representations of a fifteen
dimensional semidirect sum spectrum generating algebra $gcm(3)$ spanned by its
nonlinear $gl(3,{\bf R})$ subalgebra plus an abelian nonlinear inertia tensor
subalgebra.
