Characteristic Lie algebra of Klein-Gordon equation and higher symmetries

Consider Klein-Gordon equation written in the form $u_{xy}=f(u)$. We define characteristic Lie algebra $\chi(f)$ as a Lie subalgebra in the Lie algebra of differential operators generated by two operators
$$ X_0=\frac{\partial}{\partial u}, X_f= f\frac{\partial}{\partial u_1}+D(f)\frac{\partial}{\partial u_2}+\dots+D^{n-1}(f)\frac{\partial}{\partial u_n}+\dots, $$
where $D=u_1\frac{\partial}{\partial u}+u_2\frac{\partial}{\partial u_1}+\dots+u_{n+1}\frac{\partial}{\partial u_n}+\dots$ The properties of the characteristic Lie algebra $\chi(f)$ are related to the integrability of Klein-Gordon equation. We are going to discuss characteristic Lie algebras of two integrable cases: sine-Gordon $f(u)=\sinh{h}$ equation and Tzitzeica $f(u)=e^u+e^{2u}$ equation.