The paper is devoted to the systems of equations A(x)x˙=v(x) in real finite-dimensional phase space. The elements of the matrix A are real analytic functions, as well as the components of the vector function v. Generically, the matrix A degenerates on a certain hypersurface Γ={detA(x)=0}. Points of Γ are called singular (or impasse) points of the system. The local analytic classification for such systems at their transversal singular points is presented. Transversal singular points are defined by the conditions that Γ is regular and v is transversal to Im A.