  We construct analytical solutions describing black holes and black strings in
the Einstein $SU(N)$-non-linear sigma model in $(3+1)$ dimensions. This
construction is carried out using the maximal embbeding ansatz of $SU(2)$
together with the Euler parameterization of the $SU(N)$ group, in such a way
that the non-linear sigma model equations are automatically satisfied for
arbitrary values of the flavor number $N$ while the Einstein equations can be
solved analytically. In particular, we construct black holes with spherical and
flat horizons as well as black strings that present the geometry of a
three-dimensional charged Ba\~nados-Teitelboim-Zanelli black hole on the
transverse section of the string. These configurations are not trivial
embeddings of $SU(2)$ into $SU(N)$, which allow us to explicitly show the role
that the flavor number plays on the geometry and thermodynamics of the black
holes and black strings. Finally, we perform a thermal comparison between these
configurations.
