Classification of A-graded algebras with 3 generators

The notion of A-graded algebras was introduced by V.I. Arnold, who made a complete classification for the case of 3 generators [1]. Unfortunately he did not present proofs of his statements. When our group together with him started further investigations of A-graded algebras, he suggested to me first to construct and publish the proof of his classification in the way that we need for our work.