  Inspired by the works of Zagier, we study the probability measures $\nu(t)$
with support on the flat tori which are the compact orbits of the maximal
unipotent subgroup acting holomorphically on the positive orthonormal frame
bundle $\mathcal{F}({M}_D)$ of 3-dimensional hyperbolic Bianchi orbifolds
${M}_D=\mathbb{H}^3/\widetilde{\Gamma}_D$, of finite volume and with only one
cusp. Here $\Gamma_D=PSL(2, \mathcal{O})$, where $\mathcal{O}$ is the ring of
integers of an imaginary quadratic field of class number one.
