  In hyperbolic dissipative systems, the solution of the shock structure is not
always continuous and a discontinuous part (sub-shock) appears when the
velocity of the shock wave is greater than a critical value. In principle, the
sub-shock may occur when the shock velocity $s$ reaches one of the
characteristic eigenvalues of the hyperbolic system. Nevertheless, Rational
Extended Thermodynamics (ET) for a rarefied monatomic gas predicts the
sub-shock formation only when $s$ exceeds the maximum characteristic velocity
of the system evaluated in the unperturbed state $\lambda^{\max}_0$. This fact
agrees with a general theorem asserting that continuous shock structure cannot
exist for $s >\lambda^{\max}_0 $. In the present paper, first, the shock
structure is numerically analyzed on the basis of ET for a rarefied polyatomic
gas with $14$ independent fields. It is shown that, also in this case, the
shock structure is still continuous when $s$ meets characteristic velocities
except for the maximum one and therefore the sub-shock appears only when $s
>\lambda^{\max}_0 $. This example reinforces the conjecture that, the
differential systems of ET theories have the special characteristics such that
the sub-shock appears only for $s$ greater than the unperturbed maximum
characteristic velocity. However, in the second part of the paper, we construct
a counterexample of this conjecture by using a simple $2 \times 2$ hyperbolic
dissipative system which satisfies all requirements of ET. In contrast to
previous results, we show the clear sub-shock formation with a slower shock
velocity than the maximum unperturbed characteristic velocity.
