  We consider an $n\times n$ linear system of ODEs with an irregular
singularity of Poincar\'e rank 1 at $z=\infty$, holomorphically depending on
parameter $t$ within a polydisc in $\mathbb{C}^n$ centred at $t=0$. The
eigenvalues of the leading matrix at $z=\infty$ coalesce along a locus $\Delta$
contained in the polydisc, passing through $t=0$. Namely, $z=\infty$ is a
resonant irregular singularity for $t\in \Delta$. We analyse the case when the
leading matrix remains diagonalisable at $\Delta$. We discuss the existence of
fundamental matrix solutions, their asymptotics, Stokes phenomenon and
monodromy data as $t$ varies in the polydisc, and their limits for $t$ tending
to points of $\Delta$. When the deformation is isomonodromic away from
$\Delta$, it is well known that a fundamental matrix solution has singularities
at $\Delta$. When the system also has a Fuchsian singularity at $z=0$, we show
under minimal vanishing conditions on the residue matrix at $z=0$ that
isomonodromic deformations can be extended to the whole polydisc, including
$\Delta$, in such a way that the fundamental matrix solutions and the constant
monodromy data are well defined in the whole polydisc. These data can be
computed just by considering the system at fixed $t=0$. Conversely, if the
$t$-dependent system is isomonodromic in a small domain contained in the
polydisc not intersecting $\Delta$, if the entries of the Stokes matrices with
indices corresponding to coalescing eigenvalues vanish, then we show that
$\Delta$ is not a branching locus for the fundamental matrix solutions. The
importance of these results for the analytic theory of Frobenius Manifolds is
explained. An application to Painlev\'e equations is discussed.
