To the best of my knowledge, Singer never branded their machines for another company to sell.
Lots of people think they made machines for Sears though.
This is an interesting article about the sears machines, and includes a copyright infringement on Singer, that may be where the rumor started.
http://www.ismacs.net/sears/sears.html
If you can find out what the first digits are on the machine, we can probably tell you who -did- make it.
for instance 158.xxx means it was made by a company called Jaguar/Maruzen