On cylindrical minima of three-dimensional lattices

Nonzero point $\gamma=(\gamma_1,\gamma_2,\gamma_3)$ of three-dimensional lattice $\Gamma$ is called by cylindrical minimum if there exist no nonzero point $\eta=(\eta_1,\eta_2,\eta_3)$ such as
$$ \eta_1^2+\eta_2^2\le\gamma_1^2+\gamma_2^2, \quad |\eta_3|\le|\gamma_3|, \quad |\gamma|<|\eta|. $$
It is proved that the average number of cylindrical minima of three-dimensional integer lattices with determinant from $[1;N]$ is equal
$$ \mathcal C \cdot\ln N+O(1). $$