We carry out a general study of the stability of astrophysical flows that appear steady in a uniformly rotating frame .
Such a flow might correspond to a stellar pulsation mode or an accretion disk with a free global distortion giving it finite eccentricity .

We consider perturbations arbitrarily localized in the neighbourhood of unperturbed fluid streamlines .
When conditions do not vary around them , perturbations take the form of oscillatory inertial or gravity modes .
However , when conditions do vary so that a circulating fluid element is subject to periodic variations , parametric instability may occur .
For nearly circular streamlines , the dense spectra associated with inertial or gravity modes ensure that resonance conditions can always be satisfied when twice the period of circulation round a streamline falls within .

We apply our formalism to a differentially rotating disk for which the streamlines are Keplerian ellipses , with free eccentricity up to 0.7 , which do not precess in an inertial frame .
We show that for small e, the instability involves parametric excitation of two modes with azimuthal mode number differing by unity in magnitude which have a period of twice the period of variation as viewed from a circulating unperturbed fluid element .
Instability persists over a widening range of wave numbers with increasing growth rates for larger eccentricities .

The nonlinear outcome is studied in a follow up paper which indicates development of small scale subsonic turbulence .