  We describe linear and nonlinear modes in a ring-shaped waveguide with
localized gain and dissipation modeled by two Dirac $\delta$ functions located
symmetrically. The strengths of the gain and dissipation are equal, i.e., the
system obeys the parity-time symmetry. This configuration describes atomic
Bose-Einstein condensates with local loading and local elimination of atoms,
polaritonic condensates, or optical ring resonators with local pump and
absorption. We discuss the linear spectrum of such a system and show if the
location of the $\delta$ functions is slightly asymmetric, then the system can
be driven through a series of exceptional points by the change of the
gain-and-loss coefficient. In the nonlinear case, the system admits solutions
with spatially constant and periodic densities which are presented in the exact
analytical form. These solutions are supported by the current directed from the
gain potential towards the absorbing potential. The system also admits
currentless solutions. Stability and bifurcations of nonlinear solutions are
also described.
