Transition state theory (TST) approximates the reactive flux in an elementary chemical reaction by the instantaneous flux passing through a hypersurface (the "transition state") which completely divides the reactant and product regions of phase space. The rigorous classical evaluation of this instantaneous flux is carried out as a trace in phase space: effectively a multidimensional integral. We present an analysis of the momentum-space component of this flux integral for the case of a generalized reaction coordinate. The classic analysis of the canonical flux by Marcus [J. Chem. Phys. 41, 2624 (1964)] is refined by reducing the determinant which appears in the transition state partition function to a very simple form, facilitating the ensuing integration over coordinate space. We then extend the analysis to provide analytic expressions for the momentum flux integrals in both the energy-resolved, and the energy+angular-momentum-resolved microcanonical ensembles. These latter expressions allow substantial gains in the efficiency of microcanonical variational implementations of Transition State Theory with generalized reaction coordinates.