  Embedded solitons are rare self-localized nonlinear structures that,
counterintuitively, survive inside a continuous background of resonant states.
While this topic has been widely studied in nonlinear optics, it has received
almost no attention in the field of Bose-Einstein condensation. In this work,
we consider experimentally realizable Bose-Einstein condensates loaded in
one-dimensional optical lattices and demonstrate that they support continuous
families of stable three-dimensional (3D) embedded solitons. These solitons can
exist inside the resonant continuous Bloch bands because they are protected by
symmetry. The analysis of the Bogoliubov excitation spectrum as well as the
long-term evolution after random perturbations proves the robustness of these
nonlinear structures against any weak perturbation. This may open up a way for
the experimental realization of stable 3D matter-wave embedded solitons as well
as for monitoring the gap-soliton to embedded-soliton transition.
