Systems of non-linear equations, which describe excitation of electromagnetic TE and TM waves in a cylindrical waveguide by an electron beam, are shown to have the same first integrals. This leads to similarity of mechanisms of damping of wave amplitude growth. The analysis of the integrals shows that for the values R=k²_z · ω_B /  k²_⊥· δ (k_z and k_⊥ are longitudinal and transversal wave numbers, ω_B and δ are cyclotron frequency and linear increment of instability) which are greater than unity the damping of field amplitude takes place at the expense of the longitudinal velocity of particles. If R<<1 the damping of amplitude grouth takes place at the expense of change in the particle Larmor radius. Estimates of maximally accessible energy in the wave depending on the parameter value R are obtained. It is shown that the efficiency or the ratio of energy stored in the wave to the initial energy of the beam for arbitrary but not very small R is of the order (α / (α+1))·(ω / ω_B)· R^-1·a₀^-2, where ω is the wave frequency, α is the ratio of initial transversal and longitudinal beam energies, a₀=k_⊥ r_B0 , r_B0 is the initial Larmor radius of beam electrons. Under small values of R the efficiency is about of (α / (α+1))·(ω / nω_B)· (1-x²_nk / a²₀), where x_nk is the nearest to a₀ root of Bessel function of the order m or its first derivative for TM and TE waves, respectively.