  The condition for the existence of self-gravitational solitary waves (SGSWs),
and their polarity in any astrophysical compact object are theoretically found
for the first time. The pseudo-potential approach, which is valid for arbitrary
amplitude SGSWs, is emplyed. The general analytical results are applied in
white dwarfs and neutron stars to identify the basic features (polarity,
amplitude, and width) of the SGSWs formed in them. It found for the first time
that the SGSWs exist with negative self-gravitational potential in perturbed
states of white dwarfs and neutron stars. It is also estimated that for their
typical degenerate plasma parmeters, the amplitude and the width of the SGSWs
(moving with the speed 2 cm/s) in white dwarfs are $\sim -1.5$ ergs/gm and
$\sim 50$ cm, respectively, and those of the SGSWs (moving with the speed $\sim
27$ m/s) in neutron stars are $\sim -7.76\times 10^4$ Joules/kg, and $5.5$ km,
respectively.
