  We develop a theoretical framework to study the influences of coupling
asymmetry on the thermoelectrics of a strongly coupled SU($N$) Kondo impurity
based on a local Fermi liquid theory. Applying non-equilibrium Keldysh
formalism, we investigate charge current driven by the voltage bias and
temperature gradient in the strong coupling regime of an asymmetrically coupled
SU($N$) quantum impurity. The thermoelectric characterizations are made via
non-linear Seebeck effects. We demonstrate that the beyond particle-hole (PH)
symmetric SU($N$) Kondo variants are highly desirable with respect to the
corresponding PH symmetric setups in order to have significantly improved
thermoelectric performance. The greatly enhanced Seebeck coefficients by
tailoring the coupling asymmetry of beyond PH symmetric SU($N$) Kondo effects
are explored. Apart from presenting the analytical expressions of asymmetry
dependent transport coefficients for general SU($N$) Kondo effects, we make a
close connection of our findings with the experimentally studied SU(2) and
SU(4) Kondo effects in quantum dot nano structures. Seebeck effects associated
with the theoretically proposed SU(3) Kondo effects are discussed in detail.
