Thank you Eric,
had gone through all the instructins in the background posted by you. Theoretically, according to Google notes, with System 10.3.3. I shoild not need two step verification.
Doing sign-in of google account with no 2-step verification, i received an empty window with an endlessly rotating circle. and then an advice (in the Force Quit procedure) of MAIL not responding.
Doing the circuit via two step verificaction, I went though all the steps outlined in the Google instructions (see windows) and finally got a window where I was asked to mark which functions the mail application should manage, of which I selected some (incl. mail). But finally, after pressing "accept", the mail app. went into freeze again. Any other ideas ?
Many thanks in advance.