  We consider a differentially rotating flow of an incompressible electrically
conducting and viscous fluid subject to an external axial magnetic field and to
an azimuthal magnetic field that is allowed to be generated by a combination of
an axial electric current external to the fluid and electrical currents in the
fluid itself. In this setting we derive an extended version of the celebrated
Hain-Lust differential equation for the radial Lagrangian displacement that
incorporates the effects of the axial and azimuthal magnetic fields,
differential rotation, viscosity, and electrical resistivity. We apply the
Wentzel-Kramers-Brillouin method to the extended Hain-Lust equation and derive
a new comprehensive dispersion relation for the local stability analysis of the
flow to three-dimensional disturbances. We confirm that in the limit of low
magnetic Prandtl numbers, in which the ratio of the viscosity to the magnetic
diffusivity is vanishing, the rotating flows with radial distributions of the
angular velocity beyond the Liu limit, become unstable subject to a wide
variety of the azimuthal magnetic fields, and so is the Keplerian flow. In the
analysis of the dispersion relation we find an evidence of a new
long-wavelength instability which is caught also by the numerical solution of
the boundary value problem for a magnetized Taylor-Couette flow.
