Just a follow-up, after receiving the URL I confirmed that is a problem (the token.php is returning a weird error).
As I said in my original post, the underlying technology of the desktop version will make the desktop app stop working because it expects a JSON response and not a standard Moodle error (and it's nothing we can't control)
This was recently reported in MOBILE-2375
So, in order to fix this, you need to go the Moodle FAQ, https://docs.moodle.org/34/en/Moodle_Mobile_FAQ#I_can.27t_get_the_app_to_work._What_can_I_do.3F, and check one by one all the possible errors.
In this case it seems you are using an authentication plugin badly configured (or a non-core authentication plugin), so check points 2 and 3.