The authors ask whether it is possible to obtain Sturm-Liouville problems (SLPs) with transmission conditions having a finite spectrum of eigenvalues. They consider the (SLP) given by \[ (-px')'+qx=\lambda w x \quad \text{ on } J=(a,c)\cup (c,b),~ -\infty 0, \det(D)>0;\) \(\lambda\) is the spectral parameter; and the coefficients \(r=\frac{1}{p}, q, w\) are complex valued functions which are Lebesgue integrable on \(J.\) The answer is in affirmative, in fact, they prove (Corollary 3.2): ``For every \(m\in\{1,2,\dots\}\) there exist a (SLP) (as described above) with piecewise constant coefficients which has exactly \(m\) eigenvalues. Furthermore, given any \(k\) disjoint open sets \(\mathcal{N}_i\) in the complex plane and any \(k\) integers \(n_i\), there exists a (SLP) with transmission conditions with piecewise constant coefficients having exactly \(n_i\) eigenvalues in \(\mathcal{N}_i\) for \(i=1,\dots,k.\) Given any \(k\) disjoint open real intervals \(J_i\) and any \(k\) integers \(n_i\) there exists a self-adjoint (SLP) with transmission conditions with piecewise constant coefficients having exactly \(n_i\) eigenvalues in the intervals \(J_i\), \(i=1,\dots,k\).'' The construction of these (SLPs) follows the strategy developed in [\textit{Q. Kong, H. Wu} and \textit{A. Zettl}, J. Math. Anal. Appl. 263, 748--762 (2001; Zbl 1001.34019)].