  We present a model-independent parametrization of the $B^\pm\to J/\psi
K^\pm$, $B_d\to J/\psi K_S$ decay amplitudes by taking into account the
constraints that are implied by the isospin symmetry of strong interactions.
Employing estimates borrowed from effective field theory, we explore the impact
of physics beyond the Standard Model and introduce -- in addition to the usual
mixing-induced CP asymmetry $A_{CP}^{mix}$ in $B_d\to J/\psi K_S$ -- a set of
three observables, allowing a general analysis of possible new-physics effects
in the $B\to J/\psi K$ system. Imposing a dynamical hierarchy of amplitudes, we
argue that one of these observables may already be accessible at the
first-generation $B$-factories, whereas the remaining ones will probably be
left for second-generation $B$ experiments. However, in the presence of large
rescattering effects, all three new-physics observables may be sizeable. We
also emphasize that a small value of $A_{CP}^{mix}$ could be due to new-physics
effects arising at the $B\to J/\psi K$ decay-amplitude level. In order to
establish such a scenario, the observables introduced in this paper play a key
role.
