After expanding the distribution function about an anisotropic Planck function, the new moment closure method of Banach and Larecki applied to the Boltzmann–Peierls equation for the phonon gas dynamics leads to a whole hierarchy of closed systems of moment equations. The system of equations for the energy density and the heat flux is the first, non-perturbative member of this hierarchy of closures. In our previous paper (2004 J. Phys. A: Math. Gen. 37 9805), emphasis was placed on deriving the next member, the 9-moment anisotropic closure that involves the flux of the heat flux as an extra gas-state variable. Here, as a first step in effectively analysing this system, we present a study of the one-dimensional, rotationally symmetric reduction of these equations. Under the assumption of Callaway's model, a systematic procedure is derived which shows that the obtained system of three evolution equations for three nonvanishing gas-state variables can be cast into a symmetric hyperbolic form. For the sake of completeness, we describe explicitly the region of symmetric hyperbolicity in parameter space (the space defined by the gas-state variables). The evolution system is symmetric hyperbolic for significant ranges of physical conditions, i.e., there are effectively no unphysical limitations on the magnitude of the energy density and the heat flux. This paper also deals with the eigenvalue problem and calculates approximately the characteristic speeds.