The Wick's rotation $W$ facilitates dealing with integrals in the Minkowski space by rotating time into the Euclidean space. As this rotation in time is performed within integrals, one can view that as a change of integral's variables, and subsequently, the integrated results in the Minkowski and Euclidean space are the same.
On the other hand, spatial rotations $\cal R$ can be viewed as transformations that, depending on the properties of our system of interest with Hamiltonian $H$, may play the role of symmetries such that $${\cal R} H {\cal R}^{-1} = H.$$
My questions are
Can we consider the Wick's rotation combined with spatial rotation as a spacetime transformation for a time-dependent $H$? i.e., writing $$(W{\cal R}) H (W{\cal R})^{-1}=H~? $$
If yes, can you suggest a time-dependent Hamiltonian in which the Wick's rotation is its symmetry?
Sharing further insights regarding this issue is very appreciated.
