The authors consider a one-dimensional stochastic process \(x_ t\) satisfying \(dx=dw\), \(x_ 0=x^{in}\), and the observation process \(y_ t\) satisfying \(dy=x^ 3dt+dv\), \(y_ 0=0\), where w and v are Wiener processes independent of each other and of \(x^{in}\). Using smooth functions \(F(x_ t)\) of \(x_ t\), conditional statistics of \(x_ t\) at time t may be obtained as the conditional expectation \(\hat F_ t\) of \(F(x_ t)\) given the observations \((y_ s:0\leq s\leq t)\) up to time t. The authors state that for nonconstant F there cannot exist smooth finite-dimensional recursive filters for \(\hat F_ t\), i.e. dynamical systems on a smooth finite-dimensional manifold driven by the observation process \(y_ t\) and producing \(\hat F_ t\) as an output. The system- theoretic, analytic, and Lie-algebraic parts of the proof are outlined and discussed in detail. It is pointed out where at present there are problems in generalizing the results for other nonlinear filtering problems.