Characteristic algebras and integrable exponential systems

In the present paper we study characteristic algebras for exponential systems corresponding to degenerate Cartan matrices. These systems generalize hyperbolic sine-Gordon and Tzitzeica equations well-known in the theory of integrable systems. For such systems, corresponding to Cartan matrices of rank $2$, we describe explicitly characteristic algebras in terms of generators and relations and we prove that they have linear growth. We study the relations between the higher symmetries of these systems and the structure of their characteristic algebras. We describe completely the higher symmetries of exponential systems corresponding to the Cartan matrix of affine Lie algebra $A^{(1)}_2$. We also obtain partial results on symmetries of such systems corresponding to other degenerate Cartan matrices of rank $2$. We propose a conjecture on the structure of higher symmetries of arbitrary exponential system corresponding to a degenerate Cartan matrix. We study an interesting combinatorics related to an operator generating a characteristic algebra in the simplest case for a Darboux integrable Liouville equation. The found combinatorial properties can be very useful for proving the aforementioned conjecture on the structure of higher symmetries. Moreover, in the present paper we give a rigorous meaning to the concept of a characteristic algebra of a hyperbolic system used for a long time in the literature. We do this by means of the notion of Lie-Rinehart algebra and at the examples we demonstrate that such formalization is indeed needed.