Ok, here comes a final note:
If the above equation is discretized, giving:
$ f(q_m) = \sum_n f(p_n) K(q_m, p_n) $
one can see that
$ f(q) $ needs to be an eigenvector for the matrix
$ K $ with the eigenvalue 1. So maybe there are solutions. This actually could be explored with Mathematica. (This is always true when
$ K $ were the Kronecker-Delta - in analogy to the above.)
In principal I want to support what Sean was indicating: Before applying any tool or algorithm one has to switch off the computer, take paper and pencil and think about true nature of the problem.
Henrik