I found a bit more now. This is a video showing the mechanics of a Free Sewing Machine Company sewing machine no. 255C. It is from Rockford, Illinois. I am not sure when the machine was produced.
The mechanism seems to be named "Rotoscillo". This is youtube video showing the mechanics of this machine:
https://youtu.be/gHdjd3ejKgs
This is a link to a manual:
http://ismacs.net/free/manuals/freevs-c-e-ce-manual.pdf
It have got the same kind of wobbling mechanism, but placed in another position. Furthermore it got a bearing link system for the needle bar and take-up lever as this Köhler machine got.
The machine from Free Sewing Machine Company has been presented before in this forum:
Questions about The Free Sewing machine