This paper discusses the role and the relationships among three coupled diophantine polynomial equations involved in the solution of the optimal continuous-time LQ regulation problem. The emphasis is on establishing conditions under which the LQ regulator can be obtained via the solution of a single uncoupled diophantine equation. This analysis is motivated by the fact that, although recent papers have clarified the matter for the discrete-time domain, some subtle differences exist in the continuous-time domain. As a consequence, the extension of the discrete-time results to the continuous-time context requires a deeper analysis, which is the main aim of this paper.