  Recent gravitational wave observations of binary black hole mergers and a
binary neutron star merger by LIGO and Virgo Collaborations associated with its
optical counterpart constrain deviation from General Relativity (GR) both on
strong-field regime and cosmological scales with high accuracy, and further
strong constraints are expected by near-future observations. Thus, it is
important to identify theories of modified gravity that intrinsically possess
the same solutions as in GR among a huge number of theories. We clarify the
three conditions for theories of modified gravity to allow GR solutions, i.e.,
solutions with the metric satisfying the Einstein equations in GR and the
constant profile of the scalar fields. Our analysis is quite general, as it
applies a wide class of single-/multi-field scalar-tensor theories of modified
gravity in the presence of matter component, and any spacetime geometry
including cosmological background as well as spacetime around black hole and
neutron star, for the latter of which these conditions provide a necessary
condition for no-hair theorem. The three conditions will be useful for further
constraints on modified gravity theories as they classify general theories of
modified gravity into three classes, each of which possesses i) unique GR
solutions (i.e., no-hair cases), ii) only hairy solutions (except the cases
that GR solutions are realized by cancellation between singular coupling
functions in the Euler-Lagrange equations), and iii) both GR and hairy
solutions, for the last of which one of the two solutions may be selected
dynamically.
