The Sun does wobble, and the Centre of mass of the Solar System is not in the centre of the sun but at a point away some distance it.
The Planets Jupiter and Saturn have an effect on the Sun more so when they are both on the same side as each other.
There is an hypothasis that the Sun has a companion, by the way which I believe in.
Here is a site that you might get some info on your question.
http://www.binaryresearchinstitute.org/index.shtml
I hope this helps
Ron