  The functional equation f(p(z))=g(q(z)) is studied, where p,q are polynomials
and f,g are trancendental meromorphic functions in C. We find all the pairs p,q
for which there exist nonconstant f,g satisfying our equation and there exist
no rational f,g with this property. In fact, a more general problem is solved.
In addition to algebraic methods, some results from local analytic dynamics are
used.
