The classical integrability of sigma models on supercoset spaces is well studied, but many complications occur when one tries to study theories that contain non-coset degrees of freedom, like superstring theories on AdS(4)xCP(3) and AdS(2)xS(2)xT(6) backgrounds. I will discuss the progress that has been made in understanding the role of the non-coset fermions in these two theories, demonstrating how these fermions enter into the structure of the Lax connection in a way that preserves its Z4 invariance. I will also show how similarities found between the two cases allowed us to obtain the geometry of the complete type IIA $AdS_2\times S^2\times T^6$ superspace to all orders in the coset fermions and to the second order in the non-coset ones.
