  Low angular momentum, general relativistic, axially symmetric accretion of
hydrodynamic fluid onto Schwarzschild black holes may undergo more than one
critical transition. To obtain the stationary integral solutions corresponding
to such multi-critical accretion flow, one needs to employ numerical solutions
of the corresponding fluid dynamics equations. In the present work, we develop
a {\it{completely analytical}} solution scheme which may be used to find
several trans-critical flow behaviours of aforementioned accretion, without
explicitly solving the flow equations numerically. We study all possible
geometric configurations of the flow profile, governed by all possible
thermodynamic equations of state. We use Sturm's chain algorithm to find out
how many physically acceptable critical points the accretion flow can have, and
discuss the transition from the mono to the multi-critical flow profile, and
related bifurcation phenomena. We thus illustrate, completely analytically, the
application of certain aspects of the dynamical systems theory in the field of
large scale astrophysical flow under the influence of strong gravity. Our work
may possibly be generalized to calculate the maximal number of equilibrium
points certain autonomous dynamical systems can have in general.
