  Quantum many-body scar (QMBS) and quantum integrability(QI) have been
recognized as two distinct mechanisms for the breakdown of eigenstate
thermalization hypothesis(ETH) in an isolated system. In this work, we reveal a
smooth route to connect these two ETH-breaking mechanisms in the Ising chain
with transverse and longitudinal fields. Specifically, starting from an initial
Ising anti-ferromagnetic state, we find that the dynamical system undergoes a
smooth non-thermal crossover from QMBS to QI by changing the Ising
coupling($J$) and longitudinal field($h$) simultaneously while keeping their
ratio fixed, which corresponds to the Rydberg Hamiltonian with an arbitrary
nearest-neighbor repulsion. Deviating from this ratio, we further identify a
continuous thermalization trajectory in ($h,J$) plane that is exactly given by
the Ising transition line, signifying an intimate relation between
thermalization and quantum critical point. Finally, we map out a completely
different dynamical phase diagram starting from an initial ferromagnetic state,
where the thermalization is shown to be equally facilitated by the resonant
spin-flip at special ratios of $J$ and $h$. By bridging QMBS and QI in Ising
chains, our results demonstrate the breakdown of ETH in much broader physical
settings, which also suggest an alternative way to characterize quantum phase
transition via thermalization in non-equilibrium dynamics.
