Does the energy of the source change when gravitational waves are emitted in Einstein's theory of gravitation?

It is shown that the total “energy” and total flux of “energy” of plane gravitational wave evaluated by means of any pseudotensor are equal to zero. The well-known (14) result by Einstein that the source energy decreases in the radiation of weak plane gravitational waves does not take place in Einstein's theory of gravitation. The examples of exact wave solutions with pseudotensor equal identically to zero are given in Einstein's theory of gravitation. The energy-momentum of any weak gravitational wave is equal to zero. When such waves are radiated, the source energy cannot change although the waves are the real waves of curvature. Thus, in the essence, in Einstein's theory of gravitation the production of energy out of pure nothing takes place.