Bilinear integrals of the radiative transfer equation

It is shown that the group of problems in the theory of radiative transfer that are reducible to the sourcefree problem admits a class of integrals involving quadratic moments of the intensity of arbitrarily high orders. Based on a variational principle, it is found that these integrals, which include the R-integral, follow from the corresponding conservation laws. Some of the results are generalized to the case of anisotropic scattering.