When considering a linear-Gaussian state space model, it is often referred that, optimal inference is tractable which is very rare in state space models. When considering a nonlinear state space model, it is said that, in general the posterior distribution is intractable, therefore, inference methods should be used to approximate the posterior.
What I would like to ask is that, what is the exceptions for nonlinear state space models? Which nonlinear state space model is tractable analytically? Are there any examples?
PS: For instance, Doucet says:
Given the ﬁltering distribution one can then routinely proceed to ﬁltered point estimates such as the posterior mode or mean of the state. This problem is known as the Bayesian ﬁltering problem or the optimal ﬁltering problem. [...] Except in a few special cases, including linear Gaussian state space models (Kalman ﬁlter) and hidden ﬁnite-state space Markov chains, it is impossible to evaluate these distributions analytically.
To reexpress, I would like to ask that, which nonlinear state space model is tractable?