  We consider the $\dbar-$ equation in $\C^1$ in classes of functions with
Gaussian decay at infinity. We prove that if the right-hand side of the
equation is majorated by $\exp(-q|z|^2)$, with some positive $q$, together with
derivatives up to some order, and is orthogonal, as a distribution, to all
analytical polynomials, then there exists a solution with decays, together with
derivatives, as $\exp(-q'|z|^2)$, for any $q'<q/e$. This result carries over to
the $\dbar$-equation in classes of distributions, again, with Gaussian decay at
infinity, in some precisely defined sense. The properties of the solution are
used further on to prove the finite rank theorem for Toeplitz operators with
distributional symbols in the Fock space: the symbol of such operator must be a
combination of finitely many $\delta$-distributions and their derivatives. The
latter result generalizes the recent theorem on finite rank Toeplitz operators
with symbols-functions.
