  Let $G\subset \C P^n$ be a linearly convex compact with smooth boundary,
$D={\C}P^n\setminus G$, and let $D^* \subset (\C P^n)^*$ be the dual domain.
Then for an algebraic, not necessarily reduced, complete intersection
subvariety $V$ of dimension $d$ we construct an explicit inversion formula for
the complex Radon transform $R_V:\ H^{d,d-1}(V\cap D)\to H^{1,0}(D^*)$, and
explicit formulas for solutions of an appropriate boundary value problem for
the corresponding system of differential equations with constant coefficients
on $D^*$.
