  This paper considers the Cauchy problem for the quasilinear hyperbolic system
of balance laws in $\mathbb{R}^d$, $d\ge 2$. The system is partially
dissipative in the sense that there is an eigen-family violating the Kawashima
condition. By imposing certain supplementary degeneracy conditions with respect
to the non-dissipative eigen-family, global unique smooth solutions near
constant equilibria are constructed. The proof is based on the introduction of
the partially normalized coordinates, a delicate structural analysis, a family
of scaled energy estimates with minimum fractional derivative counts and a
refined decay estimates of the dissipative components of the solution.
