  Borel summable divergent series usually appear when studying solutions of
analytic ODE near a multiple singular point. Their sum, uniquely defined in
certain sectors of the complex plane, is obtained via the Borel--Laplace
transformation. This article shows how to generalize the Borel--Laplace
transformation in order to investigate bounded solutions of parameter dependent
non-linear differential systems with two simple (regular) singular points
unfolding a double (irregular) singularity. We construct parametric solutions
on domains attached to both singularities, that converge locally uniformly to
the sectoral Borel sums. Our approach provides a unified treatment for all
values of the complex parameter.
