  In the present work, we investigate a cut finite element method for the
parameterized system of second-order equations stemming from the splitting
approach of a fourth order nonlinear geometrical PDE, namely the Cahn-Hilliard
system. We manage to tackle the instability issues of such methods whenever
strong nonlinearities appear and to utilize their flexibility of the fixed
background geometry -- and mesh -- characteristic, through which, one can avoid
e.g. in parametrized geometries the remeshing on the full order level, as well
as, transformations to reference geometries on the reduced level. As a final
goal, we manage to find an efficient global, concerning the geometrical
manifold, and independent of geometrical changes, reduced order basis. The
POD-Galerkin approach exhibits its strength even with pseudo-random
discontinuous initial data verified by numerical experiments.
