We describe a multivariate generalisation of skew-polynomials that does not restrict the commutation relations among the generators or between the generators and the scalars. Our operator rings are thus general enough to emcompass both Ore algebras and PBW-type algebras, while allowing for non-commutative associated graded rings. As application of such rings, we describe an algorithm for computing spans in modules over them. That algorithm generalizes the cyclic vector construction to arbitrary partial functional equations, and also answers the problem of computing solutions of systems whose selected entries are in a given closed-form class of functions.