[en] The basic assumptions concerning the Quasiparticle-Phonon Nuclear Model are formulated and the mathematical apparatus is developed. The Hamiltonian, containing a finite-rank separable isoscalar and isovector multipole, a spin-multipole and a tensor particle-hole as well as particle-particle interactions transforms to a form containing quasiparticle, phonon and quasiparticle-phonon interactions. The general RPA equation is derived and the particular cases are discussed. The very complex interaction does not complicate the description of the fragmentation one-phonon states. It is shown that the three-phonon terms added to the one- and two-phonon terms in the wave function lead to an additional small shift of the two-phonon poles in the secular equation. The influence of the density-dependent separable interaction on the vibrational states is small. A common description of the collective, weakly collective and two-quasiparticle states in doubly-even well-deformed nuclei is obtained. (author)