A class of integrable models of (1 + 1)-dimensional dilaton gravity coupled to scalar and electromagnetic fields is obtained and explicitly solved. More general models are reduced to (0 + 1)-dimensional Hamiltonian systems, for which two integrable classes (called s-integrable) are found and explicitly solved. As a special case, static spherical solutions of the Einstein gravity coupled to electromagnetic and scalar fields in any real spacetime dimension are derived. A generalization of the “no-hair” theorem is pointed out and the Hamiltonian formulation that enables an exact quantization of the s-integrable systems is outlined.

Starting from the Lagrangian formulation of the Einstein equations for the vacuum static spherically symmetric metric, we develop a canonical formalism in the radial variable r that is time-like inside the Schwarzschild horizon. The Schwarzschild mass turns out to be represented by a canonical function that commutes with the r-Hamiltonian. We investigate the Wheeler-DeWitt quantization and give the general representation for the solution as superposition of eigenfunctions of the mass operator.

Paragrassmann algebras with one and many paragrassmann variables are considered from the algebraic point of view without using the Green ansatz. A differential operator with respect to paragrassmann variable and a covariant para-super-derivative are introduced giving a natural generalization of the Grassmann calculus to a paragrassmann one. Deep relations between paragrassmann algebras and quantum groups with deformation parameters being roots of unity are established.

We describe a class of explicitly integrable models of (1+1)-dimensional dilaton gravity coupled to scalar fields in sufficient detail. The equations of motion of these models reduce to systems of Liouville equations with energy and momentum constraints. We construct the general solution of the equations and constraints in terms of chiral moduli fields explicitly and briefly discuss some extensions of the basic integrable model. These models can be related to higher-dimensional supergravity theories, but we mostly consider them independently of such interpretations. We also briefly review other integrable models of two-dimensional dilaton gravity.