The Cartesian representation of the Taylor-Couette system in the vanishing gap limit by Nagata (2023) is applied to the case of counter-rotating cylinders. We find that the critical state for the onset of axisymmetric instability is determined by the product of Q and R only, where Q results from the curvature term in the radial momentum equation and R is the Reynolds number. This means that although our system is reduced to that of plane Couette flow as Q approaches zero, the critical state can still exist as R goes to infinity. After confirming this by a linear stability analysis, we proceed to obtain nonlinear axisymmetric flows. It is found that the nonlinear states exist super-critically, and that their mean flows are characterized by asymmetric profiles across the cylinder gap.