  Finite-dimensional Lie algebras of vector fields determine geometrical
collective models in quantum and classical physics. Every set of vector fields
on Euclidean space that generates the Lie algebra sl(3, R) and contains the
angular momentum algebra so(3) is determined. The subset of divergence-free
sl(3, R) vector fields is proven to be indexed by a real number $\Lambda$. The
$\Lambda=0$ solution is the linear representation that corresponds to the
Riemann ellipsoidal model. The nonlinear group action on Euclidean space
transforms a certain family of deformed droplets among themselves. For positive
$\Lambda$, the droplets have a neck that becomes more pronounced as $\Lambda$
increases; for negative $\Lambda$, the droplets contain a spherical bubble of
radius $|\Lambda|^{{1/3}}$. The nonlinear vector field algebra is extended to
the nonlinear general collective motion algebra gcm(3) which includes the
inertia tensor. The quantum algebraic models of nonlinear nuclear collective
motion are given by irreducible unitary representations of the nonlinear gcm(3)
Lie algebra. These representations model fissioning isotopes ($\Lambda>0$) and
bubble and two-fluid nuclei ($\Lambda<0$).
