I'm asked to consider the 1-param. group of transformations generated by $V = \dfrac{\partial}{\partial u} + \alpha t \dfrac{\partial}{\partial x}$, which easily enough yields $g^{\epsilon}(x,t,u) = (x+\alpha t \epsilon, t, u+\epsilon)$.
Then I'm asked to find $\alpha$ s.t. $V$ generates a Lie point symmetry of $\Delta[x,t,u] = u_t + u_{xxx} - 6uu_x$, which is where I come unstuck.
I wrote $g^{\epsilon} = (\tilde{x}, \tilde{t}, \tilde{u})$ and attempted to compute $\Delta[\tilde{x}, \tilde{t}, \tilde{u}]$, but I continually obtain results such as $\tilde{u}_{\tilde{t}} = -\alpha \epsilon u_x + u_t$, which do not seem correct (given the presence of the parameter epsilon).
Any ideas on the best way to find $\alpha$?
Thanks in advance!