From what I understand, the cocompact solvable subgroup should be the solvable radical, am I right? Then I have some difficulties concerning the statement: the unipotent radical of the Zariski closure should be trivial (here is my inexpertise, unfortunately). The rest is somewhat clear, since we are complexifying, the representation is either 1 or 2 dimensional (linked to the fact that the eigenvalues are complex or real, and the Lie Kolchin theorem), and then one studies the orbits (I believe we identify the projective line with the circle). This is what I understand, sorry for the mess.

Thanks for your reply. I've awarded the bounty to your answer already, but I would like to take some time before accepting it, because I would like to check myself your explanation. This will take me some time, since I'm not an expert. If you feel like adding some more details, please go ahead, that would be greatly appreciated. Thank you again! -Guido-

@user120527: I've added some more clarifications in my post. Thank for the reference anyway.

@FedorPetrov Yes, indeed this is what I meant, thank you for the clarification.

@YCor as you wish, but yours is the closest to an answer this post of mine has received, and much more than a wonderful starting point to begin with. So in any case thank you very much

Dear @YCor, thanks for your kind replies. Indeed a part of my question was to decide on the irreducibility of such reps, which now seems to be quinte an exceptional case. Now, do you know if there are similar (combinatorical) techniques to those discussed in the post of prof. Montgomery (e.g. with Tableaux), to describe how to build such reps? BTW if you would like to collect your comments in an answer, I will accept it and give you the bounty. Bests, Guido

@DeaneYang Nope... they do not play any particular role.

@RobertBryant I've specified which vector field $X$ I am concerned with. However your answer already contains an interesting point about existence of solution. May I ask for any particular reference that you have in mind? Best regards.

Thank you very much. What about the inhomogeneous problem $Lu=f$ on $U$?