The Pawsey Balun (or Pawsey Stub) is described as a device for connecting an unbalanced feed to a balanced antenna.
Above is a diagram of a Pawsey Balun used with a half wave dipole (ARRL).
Pawsey Balun on an asymmetric load reported model results in an asymetric dipole antenna, and showed very high common mode feed line current.
Pawsey Balun on an asymmetric load – bench load simulation showed that although the Pawsey balun is not of itself an effective voltage balun or current balun, it can be augmented to be one or the other.
So, you might ask what they do, what they are good for, and why they are used.
If you were to construct a quite symmetric half wave dipole and directly connected a coax transmission line to the centre, you would destroy the symmetry of the system as connection of the shield to one dipole leg only effectively connects the common mode conductor (the outer surface of the shield) to one leg of the dipole.
The Pawsey stub or balun is a narrowband device (ie tuned) that adapts the coaxial feed line to a pair of symmetric terminals for attachment to the antenna feed point.
In a perfectly symmetric system (source, feedline type and topology, antenna), current in the radiator will be symmetric and there will be negligible common mode current on the feed line.
Symmetry is easier to achieve with some types of VHF/UHF/SHF antennas than at HF.
Equivalent circuit of an antenna system gives measurements of a fairly symmetric G5RV Inverted V dipole + feed line and in that case, the Z1 and Z2 values are different on the two bands reported, more so on 80m.
On the other hand, a corner reflector with half wave dipole feed for 1296MHz can be constructed with very good symmetry, and fed from behind the reflector, a Pawsey balun should give the necessary feed symmetry to preserve system symmetry and have symmetric dipole currents and negligible common mode current on the feed assembly.
The question of why are they used is more difficult than the other questions. They do have application, but they are also used inappropriately and given that it is most unusual to seem validation of balun performance by measurement, such use highlights the bliss of ignorance.