How can a synchronous motor have RPM other than 3000/3600?
this motor has a reduction gear that slows it down, to 5 rpm at 50Hz or 6rpm at 60 Hz.

Actually the rotation speed of a synchronous motor in rpm is given by:
$$\frac{2 \times 60 \times  f}  n$$
where
f = frequency (Hz)
n = number of poles.

The motor in your photo cannot have 1200 poles, so it surely has a stepdown gear.