  The conditionally Markov (CM) sequence contains different classes, including
Markov, reciprocal, and so-called $CM_L$ and $CM_F$ (two CM classes defined in
our previous work). Markov sequences are special reciprocal sequences, and
reciprocal sequences are special $CM_L$ and $CM_F$ sequences. Each class has
its own forward and backward dynamic models. The evolution of a CM sequence can
be described by different models. For a given problem, a model in a specific
form is desired or needed, or one model can be easier to apply and better than
another. Therefore, it is important to study the relationship between different
models and to obtain one model from another. This paper studies this topic for
models of nonsingular Gaussian (NG) $CM_L$, $CM_F$, reciprocal, and Markov
sequences. Two models are \textit{probabilistically equivalent (PE)} if their
stochastic sequences have the same distribution, and are \textit{algebraically
equivalent (AE)} if their stochastic sequences are path-wise identical. A
unified approach is presented to obtain an AE forward/backward
$CM_L$/$CM_F$/reciprocal/Markov model from another such model. As a special
case, a backward Markov model AE to a forward Markov model is obtained. While
existing results are restricted to models with nonsingular state transition
matrices, our approach is not. In addition, a simple approach is presented for
studying and determining Markov models whose sequences share the same
reciprocal/$CM_L$ model.
