We consider the existence of a unique solution to the systems of equations that arise when we apply a Runge-Kutta method to a stiff nonlinear system of differential equations \(U'=f(t,U)\), with f satisfying a one-sided Lipschitz condition with constant \(\beta\). For any given product \(\beta\) h, where h denotes the step size, we present algebraic conditions on the Runge-Kutta matrix A which are necessary and sufficient for unique solvability of the equations. As a second topic, we consider the question whether the solution to the system of equations is stable with respect to perturbations (known as BSI-stability). For this property also, necessary and sufficient conditions on A are presented.