= Solution
A <vector> field $V=V_1\partial_x+\phi\partial_u$ is a Lie symmetry when
$$
\operatorname{pr}^{(N)}V(\Delta)=0
$$
on the equation manifold $\Delta=0$. With total <derivative> $D_x$, the prolongation coefficients obey
$$
\phi_0=\phi,\qquad
\phi_{j+1}=D_x\phi_j-u^{(j+1)}D_xV_1.
$$
For $V_1=f(x)$, direct iteration gives
$$
\begin{aligned}
\phi_3={}&\phi_{xxx}+u'(3\phi_{xxu}-f''')+3(u')^2\phi_{xuu}
+(u')^3\phi_{uuu}\\
&+3(\phi_{xu}-f''+u'\phi_{uu})u''+(\phi_u-3f')u'''.
\end{aligned}
$$
Thus $\alpha=\beta=3$.
For $u'''=u^{-3}$, invariance requires $\phi_3+3u^{-4}\phi=0$ after substituting $u'''=u^{-3}$. Equating coefficients of the independent jet variables gives
$$
\phi_{uu}=0,\qquad \phi_{xu}=f'',\qquad f'''=0,
$$
and the remaining terms force $\phi=(3a/4)u$ and $f=ax+b$. Hence the Lie algebra is spanned by
$$
\boxed{\partial_x,
\qquad
4x\partial_x+3u\partial_u.}
$$
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