  In the present paper we consider multi-scalar extension of
Einstein-Gauss-Bonnet gravity. We focus on multi-scalar Einstein-Gauss-Bonnet
models whose target space is a three-dimensional maximally symmetric space,
namely either $\mathbb{S}^3$, $\mathbb{H}^3$ or $\mathbb{R}^3$, and in the case
when the map $\text{\it spacetime} \to \text{\it target space}$ is nontrivial.
We prove numerically the existence of black holes in this class of models for
several Gauss-Bonnet coupling functions, including the case of scalarization.
We also perform systematic study of a variety of black hole characteristics and
the space-time around them, such as the area of the horizon, the entropy and
the radius of the photon sphere. One of the most important properties of the
obtained solutions is that the scalar charge is zero and thus the scalar dipole
radiation is suppressed which leads to much weaker observational constraints
compared to the majority of modified theories possessing a scalar degree of
freedom. For one of the coupling functions we could find branches of scalarized
black holes which have a nontrivial structure -- there is non-uniqueness of the
scalarized solutions belonging to a single branch and there is a region of the
parameter space where most probably stable scalarized black holes coexist with
the stable Schwarzschild black holes. Such a phenomena can have a clear
observational signature.
