  Phase conjugation with stimulated Brillouin scattering can be used to correct
wavefront aberrations in high-power laser systems. Here, we consider such
process with cylindrical vector lasers beams. By using a vectorial approach and
a modal decomposition, we obtain a differential matrix equation that enables
the calculation of the structure of Stokes vector eigenmodes and their
Brillouin gain. The emphasis is put on the mode with the largest gain, which
contributes the most to the Stokes beam. We show that the phase conjugation of
orbital angular momentum in cylindrical vector beams is prevented from
happening almost everywhere on the higher-order Poincar\'e sphere. This
phenomenon is traced to the tendency of the Stokes beam to acquire a
polarization structure similar to the incident pump beam, which constraints the
possible combinations of topological charges of the two orbital angular
momentum modes making up the Stokes cylindrical vector beams. However, near the
poles of the higher-order Poincar\'e sphere, the Stokes mode with the largest
gain acquires an additional helical phase factor that produces the conjugated
topological charge for the stronger circularly polarized component without
affecting the polarization structure. Experimental results are found to agree
with our theoretical predictions.
