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.

13 pages. 5 figures