= Solution
The flow of $W=\phi u\partial_u+\sum\alpha_ix_i\partial_{x_i}$ is $(u,x_i)\mapsto(e^{\phi s}u,e^{\alpha_i s}x_i)$; invariants are exactly solutions of $Wf=0$. Two scaling symmetries of the PDE are
$$
W_1=2u\partial_u+x\partial_x-t\partial_t,\qquad W_2=x\partial_x+y\partial_y+t\partial_t.
$$
Invariants for $W_1$ include $y,xt,ut^2$; for $W_2$, $u,y/x,t/x$. Two common invariants are $ut/x$ and $y^2/(xt)$. Invariance under $u\partial_u+x\partial_x+\tfrac12y\partial_y$ and $\partial_y$ forces $u=xF(t)$. Substitution gives $F'=F^2$, hence $u=0$ or $u=-x/(t-C)$.
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