= Solution
Use
$$
(\partial_x+\partial_y)^3
=\partial_x^3+\partial_y^3+3\partial_x\partial_y(\partial_x+\partial_y)
$$
and differentiate the spatial equation
$$
(\partial_x^2-\partial_y^2)K=uK
$$
to eliminate the mixed derivative in the time equation. After collecting the terms involving $u$ and $u_x$, one obtains
$$
\boxed{K_t+(\partial_x+\partial_y)^3K
-3u(\partial_x+\partial_y)K=0}.
$$
Set $y=x$, use $u=-2\partial_xK(x,x,t)$ with the derivative along the diagonal, and differentiate once more in $x$. The resulting compatibility condition is
$$
\boxed{u_t+u_{xxx}-6uu_x=0}.
$$
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