**Local and global integrability of two-dimensional gravity**

**Abstract:**

Two-dimensional Gravity is proved to be an integrable model. A general
solution of the equations of motion is explicitly found without gauge
fixing. It consists of two classes of solutions: (i) constant curvature
and zero torsion surfaces (the Liouville theory) and (ii) nonconstant
curvature and nonzero torsion surfaces (new solutions). Solutions are
found locally and then extended along geodesics providing global structure
of surfaces. As an example, we consider General Relativity assuming the
space-time to be a warped product of two surfaces. In this case, General
Relativity reduces to two-dimensional gravity. The vacuum solutions
contain solutions describing many wormholes, cosmic strings, domain walls
of curvature singularities, cosmic strings surrounded by domain walls,
solutions with closed timelike curves and other solutions along with
the Schwarzschild and (anti-) de Sitter solutions.