  To every Darboux integrable system there is an associated Lie group $G$ which
is a fundamental invariant of the system and which we call the Vessiot group.
This article shows that solving the Cauchy problem for a Darboux integrable
partial differential equation can be reduced to solving an equation of Lie type
for the Vessiot group $G$. If the Vessiot group $G$ is solvable then the Cauchy
problem can be solved by quadratures. This allows us to give explicit integral
formulas, similar to the well known d'Alembert's formula for the wave equation,
to the initial value problem with generic non-characteristic initial data.
