An eigenvalue problem for k Sturm–Liouville equations coupled by k parameters λ1,…,λk is considered. In contrast to the standard case, for each r, the second-order derivative in the rth equation is multiplied by λr. This problem presents various interesting features. For example, the existence of eigenvalues with oscillation counts beyond a certain (computable) value is obtained without any of the restrictive definiteness conditions known from the standard case. Uniqueness is also analysed, and the results are given greater precision via eigencurve methods in the case of two equations coupled by two parameters.