In this paper, first a detailed derivation of the Singular Integral Equation (SIE) method for application to finlines on isotropic substrates is given. The formulation is largely based on the one given by R Mittra et al. [1] for the microstrip line. The derivation is validated by computing the propagation constants of the dominant and higher order modes of bilateral finline. Next, simple perturbation formulas are given to extend the above SIE derivation to obtain the propagation constants of finlines containing anisotropic material. Numerical results are given for the propagation constants of finlines containing either an anisotropic dielectric substrate or a magnetized ferrite.