  In this present communication the integrable modulation problem has been
applied to study parametric extension of the Kapitza rotating shaft problem,
which is a protypical example of curl force as formulated by Berry and Shukla
in [J. Phys. A 45 305201 (2012)] associated with simple saddle potential. The
integrable modulation problems yield parametric time dependent integrable
systems. The Hamiltonian and first integrals of the linear and nonlinear
parametric Kapitza equation (PKE) associated with simple and monkey saddle
potentials have been given. The construction has been illustrated by choosing
$\omega(t) = a + b \cos t$ and that maps to Mathieu type equations, which yield
Mathieu extension of PKE. We study the dynamics of these equations. The most
interesting finding is the mixed mode of particle trapping and escaping via the
heteroclinic orbits depicted with the parametric Mathieu-Kapitza equation which
are absent in the case of non parametric cases.
