We consider a general pairwise Markov Gaussian linear system (X,Y), where X is hidden and Y is observed, in which an exact Kalman filter (KF) is workable. There are two kinds of particular cases: either X is Markov and Y is not, or vice versa. We show that when the processed data suit the general model, the KF based on both particular cases produce similar approximate results. This is of importance when introducing stochastic Markovian switches. In fact, it is well known that the KF is no longer workable in the first case, while it is, as detailed in the paper, in the second one