The measurement results on the thermal conductivity K(T) of solid Kr 1-x (CH 4 ) x solutions in the interval 2-40 K, x = 0-1, are reported. The temperature dependences K(T) taken in the solutions in concentrated orientationally-ordered and orientationally-disordered phases have features that are not typical of dielectric crystals. It is shown that the anomalous K(T) is caused by strong single-phonon resonance scattering at a weakly hindered quantum rotor-the CH 4 molecule with the nuclear spin I = 1.1 Introduction. A molecule freely rotating in a crystal does not interact with its surroundings. In actual practice, however, when certain properties of condensed phases containing rotating molecules (rotors) are considered, it is important to take into account even weak noncentral interactions of these molecules with the surrounding particles. The rotor-lattice interaction in a crystalline system is called a phonon-rotation interaction (PRI) [1]. Precisely PRI brings the translational vibrations of the lattice and the rotation of the molecules to equilibrium. PRI can in particular affect the thermal conductivity of the substance [2][3][4]. In the crystalline state the character of molecular rotation is dependent not only on the force of interparticle interactions but on the thermal vibrations of the lattice as well.In the solid Kr 1-x (CH 4 ) x solution we can observe quantum-mechanical rotation at low temperatures and rotational diffusion of the molecules at high temperatures. Owing to the spherical symmetry of Kr atoms and the close Lennard-Jones potential of methane and krypton, the rotation of CH 4 molecules is weakly hindered, no considerable stress are induced in the lattice of solid solution and its components are characterized by high mutual solubility in the solid phase.This investigation of thermal conductivity was made on a two-component crystal system of weakly hindered rotors and inert gas atoms whose parameters of the Lennard-Jones potential were close to those of the rotors.The goal of this work was to investigate the influence of PRI on the thermal conductivity of solid CH 4 -Kr solutions in which the CH 4 subsystem is viewed as an ensemble of rotors interacting with one to another and with the translational vibrations of the lattice. Varying the CH 4 concentration in the solution, it is possible to trace how the PRI changes on transition from the system with a weak noncentral interaction to the system with a party orientational order in orientationally-ordered phase II of methane. Three fourths of the CH 4 molecules perform heavily hindered rotational motion (small-angle orientational vibrations-librations) in a strong anisotropic molecular field of D 2d symmetry; one fourth of the molecules (rotors) rotate almost freely in the crystal field having the octahedral symmetry O h . Each of these types of rotational excitations scatters the phonons and contributes to the thermal resistance. It is found [5],