= Solution
Seek a separated solution $u=X(x)Y(y)$. On a region where the factors do not vanish, the <partial differential equation> separates as
$$
\frac{X'}X=-(6-2y)\frac{Y'}Y=\lambda.
$$
The resulting <ordinary differential equations> integrate to $X=Ae^{\lambda x}$ and $Y=B|3-y|^{\lambda/2}$ on either side of $y=3$. Choose $\lambda=2$ and the signed linear branch $Y=3-y$, which extends smoothly across that line. This yields the nonconstant solution
$$
\boxed{u(x,y)=e^{2x}(3-y)}.
$$
Direct <partial differentiation> gives $u_x=2e^{2x}(3-y)$ and $u_y=-e^{2x}$, whose prescribed combination vanishes everywhere. This direct check also covers the line on which the separated factor is zero.
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