Lax pairs for higher-dimensional evolution PDEs and a 3+1 dimensional integrable generalization of the Burgers equation

We construct Lax pairs for general $d+1$ dimensional evolution equations in the form $u_t=F[u]$, where $F[u]$ depends on the field $u$ and its space derivatives. As an example we study a $3+1$ dimensional integrable generalization of the Burgers equation. We develop a procedure to generate some exact solutions of this equation, based on a class of discrete symmetries of the Darboux transformation type. In the one-dimensional limit, these symmetries reduce to the Cole-Hopf substitution for the Burgers equation. It is discussed how the technique can be used to construct exact solutions for higher-dimensional evolution PDEs in a broader context.