The linear stability properties of an incompressible axisymmetrical vortex of axial velocity
*W*_{0}(*r*)
and angular velocity
Ω_{0}(*r*)
are considered in the limit of large Reynolds number. Inviscid approximations and viscous WKBJ approximations for three-dimensional linear normal modes are first constructed. They are then shown to be singular at the critical points *r*_{c} satisfying
ω=*m*Ω_{0}(*r*_{c}) +*kW*_{0}(*r*_{c})
, where ω is the frequency, *k* and *m* the axial and azimuthal wavenumbers. The goal of this paper is to resolve these singularities. We show that a viscous critical-layer analysis is analytically tractable. It leads to a single sixth-order equation for the perturbation pressure. This equation is identical to the one obtained in stratified shear flows for a Prandtl number equal to 1. Integral expressions for typical solutions of this equation are provided and matched to the outer inviscid and viscous approximations in the complex plane around *r*_{c}. As for planar flows, it is proved that the critical layer solution with a balanced behavior matches a non-viscous approximation in a
4π/3
sector of the complex-plane. As a consequence, the Frobenius expansions of a non-viscous mode on each side of a critical point *r*_{c} differ by a π phase jump.