Hi, and welcome to the forum!
It looks like you have a ref. 3586, which would be a carree shaped watch. The movement is the very interesting cal. 350 (or in your case it's more likely to
be the revised version which was called cal. 1-350) . Commercially it may have been a failure for Patek Philippe as it wasn't in production for a very long
time - and if I'd have to trust some comments from community members it also had it's shares of problems to wind reliably, especially if worn by more sedentary
living owners. Technically it's very interesting although, as said, even after intensive work and the second version 1-350, dismissed....
Here's a picture of the movement:
It features a so called annular rotor, on firts thought a logical solution when searching for a slim automatic movement. Of course it makes accessing
the movement laterally fro the winding/setting stem impossible - thus Patek opted for the winding stem on the back. As it turned out winding was not very efficient
and stripping down the winding train to uni directional (version 1-350) helped - but wasn't as effective as hoped. I believe that the winding stem on the back wasn't
very comfortable as well and most likely was one major reason to dismiss the concept as well.
Here's a picture of a similar watch, courtesy of Antiquorum:
I'm sure you'll find some more info by searching for "ref. 3586" or "cal. 350" and "Patek" in the famous auction houses data bases.
Hope this helps
best regards
Suitbert