  The autonomous Duffing oscillator, and its van der Pol modification, are
known to admit time-dependent first integrals for specific values of
parameters. This corresponds to the existence of Darboux polynomials, and in
fact more can be shown: that there exist Liouvillian first integrals which do
not depend on time. They can be expressed in terms of the Gauss and Kummer
hypergeometric functions, and are neither analytic, algebraic nor meromorphic.
A criterion for this to happen in a general dynamical system is formulated as
well.
