  A non-local abstract Cauchy problem with a singular integral is studied,
which is a closed system of two evolution equations for a real-valued function
and a function-valued function. By proposing an appropriate Banach space, the
well-posedness of the evolution system is proved under some boundedness and
smoothness conditions on the coefficient functions. Furthermore, an isomorphism
is established to extend the result to a partial integro-differential equation
with singular convolution kernel, which is a generalized form of the stationary
Wigner equation. Our investigation considerably improves the understanding of
the open problem concerning the well-posedness of the stationary Wigner
equation with inflow boundary conditions.
