Given a certain PDE, we have some well-known methods for proving integrability. For example, if we have a Hamiltonian system and 'sufficiently many' first integrals, we can reduce the system to quadratures. In the infinite dimensional case it is a little more complicated, but similar techniques exist, even if they are more difficult to put into practice.
On the other hand, how would one go about showing that a PDE is not integrable? What techniques are available to us?