My students found an interesting PDE today,
$ u_{xxx}+u_{yyy}+u_{zzz}-3u_{xyz} = 0. $
Does anyone happen to know if this PDE has a name?
Incidentally, one curious solution I just found is $u(x,y,z) = e^{x+y-2z}$.
Connection with other question: in my other question "Is this a wave equation" I asked about solving a system of first order ODEs in dependent variables $u,v,w$ and dependent variables $x,y,z$. I believe that solutions of that system are likewise solutions of this system. However, the curious solution (and many others Will has elucidated) are not solutions to the first order system. The third order PDE considered in this question has additional, and from my viewpoint, extraneous solutions. Moreover, the connection with circulant matrices is not at accidental! This PDE was constructed from the left-regular representation of the group algebra of the cyclic group of order $3$. In view of this origin, I find the comments here delightful.