On solving linear homogeneous grammars generating linear languages

We investigate systems of noncommutative symbolic linear homogeneous equations, which are interpreted as linear grammars of formal languages. Such systems are solved in the form of formal power series (FPS) expressing nonterminal symbols through terminal symbols of the alphabet and considered as linear languages. Each FPS is matched by its commutative image, which is obtained under the assumption that all symbols denote commutative variables, real or complex. In this paper, we consider systems of equations that can have an infinite number of solutions, parameterized not by arbitrary numbers, but by arbitrary FPS. We estimate the number of such parameters, which gives a noncommutative analogue of the well-known fact of the theory of linear equations.