  We construct a generalisation of the $\lambda$-deformation of the Principal
Chiral Model (PCM) where we deform just a subgroup $F$ of the full symmetry
group $G$. We find that demanding Lax integrability imposes a crucial
restriction, namely that the coset $F\backslash G$ must be symmetric.
Surprisingly, we also find that (when $F$ is non-abelian) integrability
requires that the term in the action involving only the spectator fields should
have a specific $\lambda$-dependence, which is a curious modification of the
procedure expected from the known $F=G$ case. The resulting Lax connection has
a novel analytical structure, with four single poles as opposed to the two
poles of the cases of the PCM and of the standard $\lambda$-deformation. We
also explicitly work out the example of $G=SU(2)$ and $F=U(1)$, discussing its
renormalisation group flow to two loops.
