in our previous publications we have considered in the even-even nuclei only the low-lying collective negative parity states with K =0. However, there are experimental data which indicate a presence of the low-lying collective states related to the reflection-asymmetric modes which are characterized by nonzero values of K . It can be also that K is not a good quantum number if nuclei are located in a transitional region between deformed and spherical ones. A good example is $^{220}$Th [12] whose energy spectra is a challenge for the theoretical approaches. To describe the properties of the low-lying collective states related to the reflection- asymmetric collective mode and characterized by nonzero values of K in the framework of the cluster approach we should take into account intrinsic excitations of the clusters forming a nucleus under consideration. The aim of the present paper is a formulation of a simple model which gives a quantitative description of the low-lying positive and negative parity bands in the spectra of nuclei like $^{220}$Th. Our model is based on the assumption that the collective oscillations of the nuclear shape lead to the formation of the cluster- type configurations. This model is further development of the previously used approaches [7, 10, 11]. The choice of the collective coordinates and the procedure of the calculation of the poten- tial energy and of the inertia coefficients for the Hamiltonian of the model are based on the concept of the dinuclear system (DNS). The concept of the dinuclear system was first intro- duced to explain the experimental data on deep inelastic and fusion reactions. Later on it was applied to the description of the nuclear structure phenomena, like alternating parity bands, mentioned above, and superdeformed states [13, 14]. The dinuclear system ( A , Z ) consists of two fragments ( A$_{1}$ , Z$_{1}$ ) and ( A$_{2}$ , Z$_{2}$ ) with A = A$_{1}$ + A$_{2}$ and Z = Z$_{1}$ + Z$_{2}$ sticked closely together by a molecular-type nucleus-nucleus potential. The degrees of freedom needed to describe the collective excitations of such a system are related to the rotation of the DNS as a whole, to the relative motion of the DNS fragments, to the intrinsic excitations of the fragments, and to the transfer of nucleons between the DNS fragments. The latter process is described usually by means of the mass-asymmetry coordinate η = ( A$_{1}$ − A$_{2}$ ) / ( A$_{1}$ + A$_{2}$ ) or by charge-asymmetry coordinate η$_{Z}$ = ( Z$_{1}$ − Z$_{2}$ ) / ( Z$_{1}$ + Z$_{2}$ ). The idea that the motion in mass-asymmetry is responsible for the formation of reflection-asymmetric deformations of the medium mass and heavy nuclei is based on the observation which manifest the alpha- cluster dinuclear system for these nuclei. Our calculations have shown that the potential 3