Consider the partial differential equation

$$\frac{\partial u}{\partial t}=u^{n} \frac{\partial u}{\partial x}+\frac{\partial^{2 k+1} u}{\partial x^{2 k+1}}$$

where $u=u(x, t)$ and $k, n$ are non-negative integers.

(i) Find a Lie point symmetry of $(*)$ of the form

$$(x, t, u) \longrightarrow(\alpha x, \beta t, \gamma u),$$

where $(\alpha, \beta, \gamma)$ are non-zero constants, and find a vector field generating this symmetry. Find two more vector fields generating Lie point symmetries of (*) which are not of the form $(* *)$ and verify that the three vector fields you have found form a Lie algebra.

(ii) Put $(*)$ in a Hamiltonian form.