The automorphism group of each cyclic quotient X_{d,e} of the affine plane has either one or two conjugacy classes of Borel subgroups, depending on whether e^2 is congruent to 1 modulo d.
Chevalley, Le normalisateur d’un groupe de Borel , S´ eminaire Claude Chevalley, tome 1 (1956-1958), exp
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Borel subgroups of the automorphism groups of affine toric surfaces
The automorphism group of each cyclic quotient X_{d,e} of the affine plane has either one or two conjugacy classes of Borel subgroups, depending on whether e^2 is congruent to 1 modulo d.