This paper first completes an investigation started by Bell and Turpin (1981). It is shown that for a general second-order system with a drift term and for which parameter values are q = 1, m = 2, McDanell's conjecture is correct. Following this, a statement is given of all necessary conditions that must be satisfied at a junction in order for a counter-example to the conjecture to exist. These conditions prove to be consistent, indicating that it is worth continuing the search for such a counterexample