  We study the integrability properties of planar N=2 superconformal field
theories in four dimensions. We show that the spin chain associated to the
planar dilation operator of N=2 superconformal QCD fails to be integrable at
two loops. In our analysis we focus on a closed SU(2|1) sector, whose two-loop
spin chain we fix by symmetry arguments (up to a few undetermined
coefficients). It turns out that the Yang-Baxter equation for magnon scattering
is not satisfied in this sector. On the other hand, we suggest that the closed
SU(2,1|2) sector, which exists in any N=2 superconformal gauge theory, may be
integrable to all loops.
