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The inadequacy of Li\'{e}nard-Wiechert potentials is demonstrated as one ofthe examples related to the inconsistency of the conventional classicalelectrodynamics. The insufficiency of the Faraday-Maxwell concept to describethe whole electromagnetic phenomena and the incompleteness of a set ofsolutions of Maxwell equations are discussed and mathematically proved. Reasonsof the introduction of the so-called ``electrodynamics dualism concept"(simultaneous coexistence of instantaneous Newton long-range andFaraday-Maxwell short-range interactions) have been displayed. It is strictlyshown that the new concept presents itself as the direct consequence of thecomplete set of Maxwell equations and makes it possible to consider classicalelectrodynamics as a self-consistent and complete theory, devoid of inwardcontradictions. In the framework of the new approach, all main concepts ofclassical electrodynamics are reconsidered. In particular, a limited class ofmotion is revealed when accelerated charges do not radiate electromagneticfield.Comment: ReVTeX file, 24pp. Small corrections which do not have influence  results of the paper. Journal reference is adde

Action at a distance as a full-value solution of Maxwell equations: The basis and application of the separated-potentials method