  It was shown recently that many of the Gustafson integrals appear in studies
of the ${\rm SL}(2,\mathbb{R})$ spin chain models. One can hope to obtain a
generalization of the Gustafson integrals considering spin chain models with a
different symmetry group. In this paper we analyse the spin magnet with the
${\rm SL}(2,\mathbb{C})$ symmetry group in case of open and periodic boundary
conditions and derive several new integrals.
