  The pairing Hamiltonian constitutes an important approximation in many- body
systems, it is exactly soluble and quantum integrable. On the other hand, the
continuum single particle level density (CSPLD) contains information about the
continuum energy spectrum. The question whether one can use the Hamiltonian
with constant pairing strength for correlations in the continuum is still
unanswered. In this paper we generalize the Richardson exact solution for the
pairing Hamiltonian including correlations in the continuum. The resonant and
non-resonant continuum are included through the CSPLD. The resonant
correlations are made explicit by using the Cauchy theorem. Low lying states
with seniority zero and two are calculated for the even Carbon isotopes. We
conclude that energy levels can indeed be calculated with constant pairing in
the continuum using the CSPLD. It is found that the nucleus $^{24}$C is
unbound. The real and complex energy representation of the continuum is
developed and their differences are shown. The trajectory of the pair energies
in the continuum for the nucleus $^{28}$C is shown.
