  We study $\mathcal{N}=2$ theories on four-dimensional manifolds that admit a
Killing vector $v$ with isolated fixed points. It is possible to deform these
theories by coupling position-dependent background fields to the flavor current
multiplet. The partition function of the deformed theory only depends on the
value of the background scalar fields at the fixed points of $v$. For a single
adjoint hypermultiplet, the partition function becomes independent of the
supergravity as well as the flavor background if the scalars attain special
values at the fixed points. For these special values, supersymmetry at the
fixed points enhances from the Donaldson-Witten twist to the Marcus twist or
the Vafa-Witten twist of $\mathcal{N}=4$ SYM. Our results explain the recently
observed squashing independence of $\mathcal{N}=2^*$ theory on the squashed
sphere and provide a new squashing independent point. Interpreted through the
AGT-correspondence, this implies the $b$-independence of torus one-point
functions of certain local operators in Liouville/Toda CFT. The
position-dependent deformations imply relations between correlators of
partially integrated operators in any $\mathcal{N}=2$ SCFT with flavor
symmetries.
