  The frequency dependent phonon Boltzmann equation is transformed to an
integral equation over the irreducible part of the Brillouin zone. Simultaneous
diagonalization of the collision kernel of that equation and a symmetry crystal
class operator allow to obtain a spectral representation of the lattice thermal
conductivity valid at finite frequency. The method is applied to C, Si and
Mg$_2$Si to obtain the static and dynamical thermal conductivities.
