  Using mathematical formalism borrowed from dynamical systems theory, a
complete analytical investigation of the critical behaviour of the stationary
flow configuration for the low angular momentum axisymmetric black hole
accretion provides valuable insights about the nature of the phase trajectories
corresponding to the transonic accretion in the steady state, without taking
recourse to the explicit numerical solution commonly performed in the
literature to study the multi-transonic black hole accretion disc and related
astrophysical phenomena. Investigation of the accretion flow around a non
rotating black hole under the influence of various pseudo-Schwarzschild
potentials and forming different geometric configurations of the flow structure
manifests that the general profile of the parameter space divisions describing
the multi-critical accretion is roughly equivalent for various flow geometries.
However, a mere variation of the polytropic index of the flow cannot map a
critical solution from one flow geometry to the another, since the numerical
domain of the parameter space responsible to produce multi-critical accretion
does not undergo a continuous transformation in multi-dimensional parameter
space. The stationary configuration used to demonstrate the aforementioned
findings is shown to be stable under linear perturbation for all kind of flow
geometries, black hole potentials, and the corresponding equations of state
used to obtain the critical transonic solutions. Finally, the structure of the
acoustic metric corresponding to the propagation of the linear perturbation
studied are discussed for various flow geometries used.
