For each real number Λ a Lie algebra of nonlinear vector fields on
three dimensional Euclidean space is reported. Although each algebra is
mathematically isomorphic to gl(3,R), only the Λ=0 vector
fields correspond to the usual generators of the general linear group. The
Λ<0 vector fields integrate to a nonstandard action of the general
linear group; the Λ>0 case integrates to a local Lie semigroup. For
each Λ, a family of surfaces is identified that is invariant with
respect to the group or semigroup action. For positive Λ the surfaces
describe fissioning nuclei with a neck, while negative Λ 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,R) subalgebra plus an abelian nonlinear inertia tensor
subalgebra.Comment: 13 pages plus two figures(available by fax from authors by request