What mechanism causes oscillations of the solar system's orbit about the galactic plane? In a recent paper (news release here) Lisa Randall and Matthew Reece propose that a dark matter disk coinciding with the galactic plane together with the solar system's oscillations through the galactic plane could explain the 35 million year periodicity in mass extinctions.  They propose that the solar system passes through this dark matter disk in the galactic plane every 35 million years, disrupting bodies in the Oort cloud and causing some of them to collide with Earth.  An image of the proposed cycles from the news release is below.

My question is: What mechanism is responsible for this 35 million year oscillation about the galactic plane?  Is there a companion body to the sun? Are we orbiting around one of the arms of our galaxy?  Is this a well known phenomenon or are they proposing the 35 million year oscillation as well as the dark matter disk?
The cause for the oscillations perpendicular to the galactic plane is the gravity of the non-spherical mass distribution (needed for a plane Kepler ellipse) in the Milky Way. Simplified, there is a dense galactic plane. The density is not exactly known; therefore there is some uncertainty (a few million years) about the precise oscillation period. Details see this article, subsection 3.3.

The idea of a correlation of mass extinctions with this oscillation isn't new, it originates back probably to about 1970, or earlier.

"The outer solar system probably does not contain a large gas giant planet, or a small, companion star", see this press release.

We are probably not orbiting around a spiral arm.

The dark matter disk is a hypotheses, an idea to investigate. Usually only a small fraction of this kind of hypotheses can be confirmed definitively later, most of them can be ruled out after some time, some remain unsolved, some can be refined to match observations.