Did Rolex produce watched without serial numbers.

[Paulust5r4ious]

I have a Rolex OysterDate Precision handed down from Grandfather to Father to Son. Rumour has it that he bought it in his twenties, which would make it 70-80 years old. I don't have a decent Macro lens, but it looks identical to this. (you can click on the little pics to blow them up).

I can remember the watch as a child almost 50 years ago. So I have no reason to believe it is a fake. But it has no serial number. On the 'North' end it says REGISTERED DESIGN between the strap holders but there is no evidence of a serial number. The face is yellowed (with age?), but it still works smoothly. A few things which may help prove/disprove this...

The winder is rotated about 270 degrees anti-clockwise to release it.
Then you can wind it up.
Then one pull-click to change the time
Not figured how to change the date and I'm loathe to force anything.
I have no way to open the rear.

So the simple question.Did Rolex ever produce watches without serial numbers? I'm sorry if this has been asked before but I gave up after an hour of trolling.

Paul