  Fermionic dark matter (DM) with attractive self-interaction is possible to
form black holes (BH) inside the Gyr-old neutron stars (NS). Therefore by
observing such NS corresponding to their adjacent DM environments can place
bounds on DM properties, eg. DM-baryon cross section $\sigma_{\chi b}$, DM mass
$m_{\chi}$, dark coupling $\alpha_\chi$ and mediator mass $m_\phi$. In case of
isospin violation, DM couples to neutron and proton in different strengths.
Even NS is composed of protons roughly one to two percent of the total baryons,
the contribution from protons to the DM capture rate could be drastically
changed in the presence of isospin violation. We demonstrate that this effect
can be important in certain cases. On the other hand, DM-forming BH inside the
star is subject to many criteria and the underlying dynamics is rich with
interesting features. We also systematically review the relevant physics based
on the virial equation. Moreover, an accompanied python package
\texttt{dm2nsbh} to realize the mechanism is also released on the github for
other relevant research.
