  We consider a problem of finding vanishing at infinity
$C^1([0,\oo))$-solutions to non-homogeneous system of linear ODEs which has the
pole of first order at $x=0$. The resonant case where the corresponding
homogeneous problem has nontrivial solutions is of main interest. Under the
conditions that the homogeneous system is exponentially dichotomic on $[1,\oo)$
and the residue of system's operator at $x=0$ does not have eigenvalues with
real part 1, we construct the so called generalized Green function. We also
establish conditions under which the main non-homogeneous problem can be
reduced to the Noetherian one with nonzero index.
