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The companion could be a main sequence star, white dwarf, neutron star or black hole.

We detected pulsar-pulsar binaryhere several decades ago. But if one of the pulsar does not radiate towards us, we may be not able to tell it is a pulsar or a black hole.

If the companion is not a main sequence star, how to know the nature of the companion in a neutron star binary?

Of the binary systems with one single neutron star, is that possible there is another neutron star or a hidden black hole?

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For something to be possible in this universe, doesn't even have to be consistent with the physics we know. Astronomical surprises have often led to revisions in our theories. Thus, all the rare situations you described are possible.

How do we detect such evasive systems?

There is no definite answer. Based on the data we have about the system, you can get creative while doing data analysis and come up with new methods. That is one of the reasons science continues to be fun.

Maybe someone who has had some experience with the concerned matter can expand.

Also, such systems are rare due to a physical reason. Mass transfer between binary systems ensures that one of the stars get evolved earlier than the other. The other star either ends up being a low mass star and evolves accordingly in most cases. The remaining cases account for your rare description of systems.

See Algol Paradox for an interesting read.

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Could you please summarize the stellar types of the other star(non main sequence) in a neutron star binary? How did we find this kind of system generally? –  questionhang May 20 at 7:35
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