In this paper, different reactor systems are studied, being all of them combinations of ideally stirred continuous tank reactors (CSTR), whose transfer functions - quotient of Laplace transforms of the reactor output and input - are always rational functions. The inverse problem consists of obtaining the reactor or system of reactors that corresponds to a certain given rational function, assuming that it exists. Therefore, the mentioned inverse problem gives rise to the study of conditions of existence, uniqueness, and stability of the solutions. In this work, this inverse problem is analyzed and it is concluded that the solution does not always exist, if it exists, it is not necessarily unique and, even more, small changes in the given rational function can produce very important changes in the reactor obtained and so, stability does not take place as it usually occurs in many typical direct problems. Several examples are presented in the article illustrating all the above results.