Small amplitude waves in plasmas are usually described by large systems of linearized Maxwell and kinetic equations. A procedure of elimination of components and reduction of the order of the system is discussed within the geometric optics approximation. In some cases characterized by Hermitian generalized dielectric tensors ℰ describing the unreduced problem, the successive reduction procedure yields, at each step, energy conserving reduced systems that preserve the general form and first-order nature of the equations. The number of equations in the final reduced system is equal to the number of degenerate vanishing eigenvalues of ℰ. The theory is applied in the case of transverse waves, propagating along the magnetic field in plasmas with plane parallel stratification. Both cold streaming plasma and kinetic problems are considered by using the same-order reduction procedure. The cold plasma case, at low densities, becomes doubly degenerate at cyclotron resonance, reflecting mode coupling between the vacuum electromagnetic and electron cyclotron modes. Mode conversion coefficients found from the solution of the reduced system of two first-order differential equations, characterizing this case, are in an excellent agreement with the results of the numerical solutions of the full unreduced system of equations. The kinetic case is approached by viewing the plasma as consisting of many beamlets each governed by the cold fluid approximation. The problem represents a multiply degenerate situation as many beamlets are in resonance at a time. Renormalized perturbation analysis of the partially reduced system in the low-density case predicts results similar to those found in the cold plasma, with possible broadening of the resonance region. At large densities, for propagation from the lower magnetic field side, the wave is reflected and the cyclotron resonance is inaccessible. In contrast, for propagation from the high magnetic field side, the electromagnetic energy is transferred to the electrons via the mechanism of Landau cyclotron damping.