Hello,
reading http://ignorantguru.github.io/spacefm/spacefm-manual-en.html#programfiles-usr-bin-spacefm-auth ,
Perhaps you should use the spacefm executable rather than the spacefm-auth one?
You are absolutely right. When I was choosing the appropriate file in
Menu -> Settings -> Preferred Applications -> Utilities tab -> File Manager -> dropdown list - Other... ->
there was no spacefm executable, only the spacefm-auth. Now it's there. I don't know what happened or what I did.
Anyway, problem solved.
Thank you very much.