The point scatterers were introduced in middle of 1930-ies as a simple model of nuclear interactions by Bethe, Peierls and Fermi. The strict functional-analytic interpretation was suggested by Berezin and Faddeeev. One of the principal problems of the multidimensional scattering theory is the lack of exactly solvable systems. We calculate the Faddeev (complex momenta) eigenfunctions for a system of point scatterers and show that this model is useful to check some hypotheses about the behavior of the Faddeev eigenfunctions in the complex domain.