Let a(x), b(x), p(x) be formal power series in the indeterminate x over [Formula: see text] (i.e. elements of the ring [Formula: see text] of such series) such that ord a(x) = 0, ord p(x) = 1 and p(x) is embeddable into an analytic iteration group [Formula: see text] in [Formula: see text]. By a covariant embedding of the linear functional equation [Formula: see text] (for the unknown series [Formula: see text]) with respect to [Formula: see text]. In this paper we solve the system ((Co1), (Co2)) (of so-called cocycle equations) completely, describe when and how the boundary conditions (B1) and (B2) can be satisfied, and present a large class of equations (L) together with iteration groups [Formula: see text] for which there exist covariant embeddings of (L) with respect to [Formula: see text].

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We investigate the translation equation