  In a class of Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories, we derive
both vacuum and interior Schwarzschild solutions under the condition that the
derivatives of a scalar field $\phi$ with respect to the radius $r$ vanish. If
the parameter $\alpha_{\rm H}$ characterizing the deviation from Horndeski
theories approaches a non-zero constant at the center of a spherically
symmetric body, we find that the conical singularity arises at $r=0$ with the
Ricci scalar given by $R=-2\alpha_{\rm H}/r^2$. This originates from violation
of the geometrical structure of four-dimensional curvature quantities. The
conical singularity can disappear for the models in which the parameter
$\alpha_{\rm H}$ vanishes in the limit that $r \to 0$. We propose explicit
models without the conical singularity by properly designing the classical
Lagrangian in such a way that the main contribution to $\alpha_{\rm H}$ comes
from the field derivative $\phi'(r)$ around $r=0$. We show that the extension
of covariant Galileons with a diatonic coupling allows for the recovery of
general relativistic behavior inside a so-called Vainshtein radius. In this
case, both the propagation of a fifth force and the deviation from Horndeski
theories are suppressed outside a compact body in such a way that the model is
compatible with local gravity experiments inside the solar system.
