In this work, we are concerned with the stationary neutron transport Boltzmann equation (in its integral form) in a parallelepiped. Functional methods allow us to prove that the integral transport operator, which is defined in L2 space, has eigenvalues depending continuously and monotonically on geometrical and physical parameters. We show that the eigenfunctions are continuous with respect to set of the spatial variables and the optical parameters. Finally, we remark that the same results are valid if the study is carried out in the Banach space C.