  We propose and investigate a model for the coexistence of Superconductivity
(SC) and Ferroelectricity (FE) based on the dynamical symmetries $su(2)$ for
the pseudo-spin SC sector, $h(4)$ for the displaced oscillator FE sector, and
$su(2) \otimes h(4) $ for the composite system. We assume a minimal
symmetry-allowed coupling, and simplify the hamiltonian using a double mean
field approximation (DMFA). A variational coherent state (VCS) trial
wave-function is used for the ground state: the energy, and the relevant order
parameters for SC and FE are obtained. For positive sign of the SC-FE coupling
coefficient, a non-zero value of either order parameter can suppress the other
(FE polarization suppresses SC and vice versa). This gives some support to
"Matthias' Conjecture" [1964], that SC and FE tend to be mutually exclusive.
For such a Ferroelectric Superconductor we predict: a) the SC gap $\Delta $
(and $T_c $) will increase with increasing applied pressure when pressure
quenches FE as in many ferroelectrics, and b) the FE polarization will increase
with increaesing magnetic field up to $H_c $. The last result is equivalent to
the prediction of a new type of Magneto-Electric Effect in a coexistent SC-FE
material. Some discussion will be given of the relation of these results to the
cuprate superconductors.
