  This paper presents a new uniform method for studying modal companions of
superintuitionistic deductive systems and related notions, based on the
machinery of stable canonical rules. Using our method, we obtain an alternative
proof of the Blok-Esakia theorem both for logics and for rule systems, and
prove an analogue of the Dummett-Lemmon conjecture for rule systems. Since
stable canonical rules may be developed for any rule system admitting
filtration, our method generalises smoothly to richer signatures. We illustrate
this by applying our techniques to prove analogues of the Blok-Esakia theorem
(for both logics and rule systems) and of the Dummett-Lemmon conjecture (for
rule systems) in the setting of tense companions of bi-superintuitionistic
deductive systems. We also use our techniques to prove that the lattice of rule
systems (logics) extending the modal intuitionistic logic $\mathtt{KM}$ and the
lattice of rule systems (logics) extending the provability logic $\mathtt{GL}$
are isomorphic.
