Lie algebras of slow growth and Klein-Gordon equation

We discuss the notion of characteristic Lie algebra of a hyperbolic PDE. The integrability of a hyperbolic PDE is closely related to the properties of the corresponding characteristic Lie algebra $\chi$. We establish two explicit isomorphisms between characteristic Lie algebras of sinh-Gordon and Tzitzeica equations and pro-solvable Lie subalgebras of affine Kac-Moody algebras $A_1^{(1)}$ and $A_2^{(2)}$ respectively. Hence both characteristic Lie algebras are slowly linearly growing Lie algebras with average growth rates $\frac{3}{2}$ and $\frac{4}{3}$ respectively.