Consider a one-parameter group of transformations acting on $\mathbb{R}^{4}$

$$(x, y, t, u) \longrightarrow(\exp (\epsilon \alpha) x, \exp (\epsilon \beta) y, \exp (\epsilon \gamma) t, \exp (\epsilon \delta) u)$$

where $\epsilon$ is a group parameter and $(\alpha, \beta, \gamma, \delta)$ are constants.

(a) Find a vector field $W$ which generates this group.

(b) Find two independent Lie point symmetries $S_{1}$ and $S_{2}$ of the $\mathrm{PDE}$

$$\left(u_{t}-u u_{x}\right)_{x}=u_{y y}, \quad u=u(x, y, t),$$

which are of the form (1).

(c) Find three functionally-independent invariants of $S_{1}$, and do the same for $S_{2}$. Find a non-constant function $G=G(x, y, t, u)$ which is invariant under both $S_{1}$ and $S_{2}$.

(d) Explain why all the solutions of (2) that are invariant under a two-parameter group of transformations generated by vector fields

$$W=u \frac{\partial}{\partial u}+x \frac{\partial}{\partial x}+\frac{1}{2} y \frac{\partial}{\partial y}, \quad V=\frac{\partial}{\partial y},$$

are of the form $u=x F(t)$, where $F$ is a function of one variable. Find an ODE for $F$ characterising these group-invariant solutions.