The Mishna at the end of the 14th perek of יבמות gives a few examples where יבום is required and חליצה cannot be done. For example, if there are two brothers, one a חירש (deaf-mute) and one a פיקח (not a חירש) and the פיקח dies, then the חירש must perform יבום and cannot perform חליצה.
Nowadays, however, we (Ashkenazim, at least) normally always do חליצה and never יבום. So what would we do now if this case or another such case were to arise? In particular, I am interested in what would happen if the brother the חירש were married considering the modern obligation (on Ashkenazim, at least) to remain monogamous? חליצה seems to be out of the question. So does he do יבום? A חירש is unable to divorce his יבמה after the fact, so does he have to divorce his wife and then do יבום? Perhaps the יבמה simply remains unmarried for the rest of her life like an aguna (this is what I suspect is the case). If that is the case, is there room to allow יבום either in the case where the brother is married or unmarried in order to prevent that situation?