The author considers two classes of systems described by nonlinear ordinary differential and difference equations, respectively. These systems consist of two subsystems with a polynomial link map. For the systems a non-stochastic nonlinear least-squares filtering problem is posed as a fixed interval optimization problem. Using Pontryagin’s maximum principle, Riccati transformations, generalized Volterra series expansions, and Lie-algebraic considerations, the author derives conditions under which the filters may be represented by finite- dimensional differential or difference systems. In addition, comparisons to corresponding stochastic nonlinear conditional mean filtering problems are performed. By means of some examples it is shown that only in some special cases including the linear one deterministic least-squares filters and stochastic conditional mean filters are equivalent.