Motivation: if we are given
$ u_x=v_y \qquad v_x=u_y $
then it follows $u_{xx}=u_{yy}$ and $v_{xx}=v_{yy}$. If we think of $x$ as time then these are one-dimensional wave equations for $u$ and $v$.
Question: suppose $u,v,w$ are functions dependent on $x,y,z$ such that $ u_x=v_y=w_z, \qquad \& \qquad v_x=w_y=u_z, \qquad \& \qquad w_x=u_y=v_z. $ Do the conditions above imply wave equations for $u,v,w$?
Something I read seems to imply these equations produce a three-dimensional wave equation. But this seems wrong since one of the variables counting as time only leaves two independent spatial variables. For example, $u_{xx}=u_{yy}+u_{zz}$ is a two-dimensional wave equation. But, perhaps the three-dimensionality refers to the dependent variables as a triple $(u,v,w)$. Maybe there is the same wave equation for each component in $(u,v,w)$? I do not insist that $x$ be the "time" in the equation, I cannot judge from the motivating example if $x$ (or $y$) plays a special role.
Thanks in advance for your insights.