Let G1β and G2β be compact Lie groups, X1ββg1β, X2ββg2β and consider the operator \begin{equation*} L_{aq} = X_1 +
a(x_1)X_2 + q(x_1,x_2), \end{equation*} where a and q are
ultradifferentiable functions in the sense of Komatsu, and a is real-valued.
We characterize completely the global hypoellipticity and the global
solvability of Laqβ in the sense of Komatsu. For this, we present a
conjugation between Laqβ and a constant-coefficient operator that preserves
these global properties in Komatsu classes. We also present examples of
globally hypoelliptic and globally solvable operators on T1ΓS3 and S3ΓS3 in the sense of Komatsu. In
particular, we give examples of differential operators which are not globally
Cβ-solvable, but are globally solvable in Gevrey spaces.Comment: 23 pages. arXiv admin note: text overlap with arXiv:1910.01922,
arXiv:1910.0005