We find and classify all first-order conservation laws in the Stroh formalism. All possible non-semisimple degeneracies are considered. The laws are found to depend on three arbitrary analytic functions. In some instances, there is an "extra" law which is quadratic in Vu. Separable and inseparable canonical forms for the stored energy function are given for each type of degeneracy and they are used to compute the conservation laws. The existence of a real Stroh eigenvector is found to be a necessary and sufficient condition for separability. The laws themselves are stated in terms of the Stroh eigenvectors.