  Embedded solitons are exceptional modes in nonlinear-wave systems with the
propagation constant falling in the system's propagation band. An especially
challenging topic is seeking for such modes in nonlinear dynamical lattices
(discrete systems). We address this problem for a system of coupled discrete
equations modeling the light propagation in an array of tunnel-coupled
waveguides with a combination of intrinsic quadratic
(second-harmonic-generating) and cubic nonlinearities. Solutions for discrete
embedded solitons (DESs) are constructed by means of two analytical
approximations, adjusted, severally, for broad and narrow DESs, and in a
systematic form with the help of numerical calculations. DESs of several types,
including ones with identical and opposite signs of their fundamental-frequency
and second-harmonic components, are produced. In the most relevant case of
narrow DESs, the analytical approximation produces very accurate results, in
comparison with the numerical findings. In this case, the DES branch extends
from the propagation band into a semi-infinite gap as a family of regular
discrete solitons. The study of stability of DESs demonstrates that, in
addition to ones featuring the well-known property of semi-stability, there are
linearly stable DESs which are genuinely robust modes.
