We consider a triplet Markov Gaussian linear systems (X;R;Y), where X is a sequence of continuous hidden states, R is a hidden discrete sequence, Y is an observed continuous sequence, and (X;Y) is Gaussian conditionally on R. In the classical "Conditionally Gaussian Linear State-Space Model" (CGLSSM), the exact computation of the first and the second moments of the filtering distribution is a NP-hard problem. By contrast, in a recent family of triplet models called "Conditionally Markov Switching Hidden Linear Models" (CMSHLM), the exact computation of these moments can be done with complexity linear in the number of observations. In this paper, we show that it is possible to modify a given CGLSSM to obtain a quite close CMSHLM in which exact optimal filtering is possible. So we provide an alternative to classical approximative filtering techniques in CGLSSM.