Hi @Artashes
If she applies for asylum, your mother will be obliged to stay at an initial reception centre (Erstaufnahmeeinrichtung). The maximum amount of time someone can stay there is generally 18 months. The obligation to live there can be lifted for certain reasons, for example a person’s health. If your mother’s health makes it unsuitable for her to live in a reception centre, then she can request to be allowed to move out of it.
There will be restrictions on where she can live in Germany during her asylum procedure. If she is in the same area as you, she can then live with you. If she is not in the same area, then she would have to make an additional request to move to your area for family reasons. If permitted, you would then have to discuss with the Sozialamt as to whether they can contribute towards her accommodation cost (i.e. your rent costs).
Best,
Éanna