  A novel ES-BGK-based model of non-polytropic rarefied gases in the framework
of kinetic theory is presented. Key features of this model are: an internal
state density function depending only on the microscopic energy of internal
modes (avoiding the dependence on temperature seen in previous reference
studies); full compliance with the H-theorem; feasibility of the closure of the
system of moment equations based on the maximum entropy principle, following
the well-established procedure of Rational Extended Thermodynamics.
  The structure of planar shock waves in carbon dioxide (CO$_2$) obtained with
the present model is in general good agreement with that of previous results,
except for the computed internal temperature profile, which is qualitatively
different with respect to the results obtained in previous studies, showing
here a consistently monotonous behavior across the shock structure, rather than
the non monotonous behavior previously found.
