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We develop a new KAM scheme that applies to SL(2, $${{\mathbb R}}$$) cocycles with one frequency, irrespective of any Diophantine condition on the base dynamics. It gives a generalization of Dinaburg-Sinai's theorem to arbitrary frequencies: under a closeness to constant assumption, the non-Abelian part of the classical reducibility problem can always be solved for a positive measure set of parameters. [ABSTRACT FROM AUTHOR]