  In a recent paper [Z.-N. Cai, Y.-W. Fan, and R. Li. Tech Report, Institude of
Math, Peking Univeristy(2013)], it was revealed that a modified 13-moment
system taking intrinsic heat fluxes as variables, instead of the heat fluxes
along the coordinate vectors which is adopted in the classical Grad 13-moment
system, attains some additional advantages than the classical Grad 13-moment
system, particularly including that the equilibrium is turned to be the
interior point of its hyperbolicity region. The modified 13-moment system was
actually derived from the generalized Hermite expansion of the distribution
function, where the anisotropy of Hermite expansion is specified by the full
temperature tensor. We extend the method therein in this paper to high order of
generalized Hermite expansion to derive arbitrary order moment systems, and
proposed a globally hyperbolic regularization to achieve locally well-posedness
similar to the method in [Z. Cai, Y. Fan, and R. Li, Comm. Pure Appl.
Math.(online)(2013)]. Furthermore, the structure of the eigen-system of the
coefficient matrix and all characteristic waves are fully clarified. The
obtained systems provide a systematic class of hydrodynamic models as the
refined version of Euler equations, which is gradually approaching the
Boltzmann equation with increasing order of the expansion.
