  The Jackiw-Teitelboim gauge formulation of the 1+1 dimensional gravity allows
us to relate different gauge fixing conditions with integrable hierarchies of
evolution equations. We show that the equations for the Zweibein fields can be
written as a pair of time reversed evolution equations of the
reaction-diffusion type, admitting dissipative solutions. The spectral
parameter for the related Lax pair appears as the constant valued spin
connection associated with the $SO(1,1)$ gauge symmetry. Spontaneous breaking
of the non-compact symmetry and the irreversible evolution are discussed.
