A new solution of Maxwell equations for a vacuum, for wire with constant and alternating current, for the capacitor, for the sphere, etc. is presented. First it must be noted that the proof of the solution's uniqueness is based on the Law of energy conservation which is not observed (for instantaneous values) in the known solution. The solution offered: Complies with the energy conservation law in each moment of time, i.e. sets constant density of electromagnetic energy flux; Reveals phase shifting between electrical and magnetic intensities; Explains existence of energy flux along the wire that is equal to the power consumed. A detailed proof is given for interested readers. Experimental proofs of the theory are considered. Explanation is proposed for the experiments, which have not yet been explained. The work offers some technical applications of the solution obtained.