The purpose of this paper is to develop a layer potential analysis in order to show the well-posedness result of a transmission problem for the Oseen and Brinkman systems in open sets in $${\mathbb R}^m$$ R m ( $$m\in \{2,3\}$$ m ∈ { 2 , 3 } ) with compact Lipschitz boundaries and around a lower dimensional solid obstacle, when the boundary data belong to some $$L^q$$ L q -spaces. If $$m=3$$ m = 3 or if the Brinkman system is given on bounded open set then there exists a solution of the transmission problem for arbitrary data. If $$m=2$$ m = 2 and the Brinkman system is given on exterior open set then necessary and sufficient conditions for the existence of a solution of the transmission problem are stated. A solution of the transmission problem is not unique. All solutions of the problem are found.