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In this work the numerical stability of a streamline singular hyperbolic/saddle critical point (HSP) and its relationship with the divergence of pressure force/fluid flux are numerically investigated at low Reynolds numbers. Three canonical configurations at different Reynolds numbers are considered: (a) an isolated circular cylinder; (b) side-by-side circular cylinders and (c) a near-wall circular cylinder. The objective is to investigate the behavior of a HSP subjecting to imbalanced shear-layer interaction and different Reynolds numbers. It is found that a HSP evolves along the shear-layer interfaces and imposes adverse pressure gradients, which potentially deteriorates near-wake stability. The inherent characteristics of a HSP is linked with net positive fluid-fluid divergence and fluid three dimensionality. A vorticity-free stagnant zone is formed around a HSP, which cut the kinetic energy supply of shear layers in wake, project third-dimensional fluid fluxes and develops three-dimensional streamwise braids. These findings are confirmed and explained via the quantification of the fluid-flux divergence, the hydrodynamic responses of cylinder(s) and the secondary enstrophy. The primary focus in this article is to reveal the subtle analytical relationships between HSP, divergence of pressure force/fluid flux, fluid three dimensionality and imbalanced shear-layer interaction. To the knowledge of authors, so far these analytical relationships have not been reported in literature.