  We find stealth Schwarzschild solutions with a nontrivial profile of the
scalar field regular on the horizon in the Einstein gravity coupled to the
scalar field with the k-essence and/or generalized cubic galileon terms, which
is a subclass of the Horndeski theory breaking the shift symmetry, where the
propagation speed of gravitational waves coincides with the speed of light.
After deriving sufficient conditions for the shift symmetry breaking theory to
allow a general Ricci-flat metric solution with a nontrivial scalar field
profile, we focus on the stealth Schwarzschild solution with the scalar field
with or without time dependence. For the profile $\phi=\phi_0(r)$, we
explicitly obtain two types of stealth Schwarzschild solutions, one of which is
regular on the event horizon. The linear perturbation analysis clarifies that
the kinetic term of the scalar mode identically vanishes, indicating that the
scalar mode is strongly coupled. The absence of the kinetic term of the scalar
mode in the quadratic action would inevitably arise for the stealth
Schwarzschild solutions in the theory with a general scalar field profile
depending only on the spatial coordinates. On the other hand, for the
time-dependent scalar field profile, we clarify that there does not exist a
stealth Schwarzschild solution in the shift symmetry breaking theories.
