On covariant Embeddings of a Linear Functional equation with Respect to an Analytic Iteration Group

Let a(x), b(x), p(x) be formal power series in the indeterminate x over C (i.e., elements of the ring C [[x]] of such series) such that ord a(x) = 0, ord p(x) = 1 and p(x) is embeddable into an analytic iteration group (π(s, x))s∈C in C [[x]]. By a covariant embedding of the linear functional equation φ(p(x)) = a(x)φ(x) + b(x), (L) (for the unknown series φ(x) ∈ C [[x]]) with respect to (π(s, x))s∈C we understand families (α(s, x))s∈C and (β(s, x))s∈C with entire coefficient functions in s, such that the system of functional equations and boundary conditions