halacha - What do we do when chalitza is impossible?

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?

Answer

In 1724, Rabbi Yaakov Reischer was approached by a woman named Chavah in this predicament: her deceased husband's only remaining brother Yissachar had a deformity on his foot which prevented Chalitza being done. He ruled that Yibbum must be performed, and that R Gershon's prohibition on polygamy does not apply where it would leave one party unable to ever remarry^[1]. He elaborates on the text of the blessings involved, the Ketuba he used, etc. in his Shevut Yaakov 3:134-137.

This ruling is cited approvingly in Pitchei Teshuva EH 165:3. Such levirate marriages have since been prescribed by R Chaim Ozer Grodzinski (Achiezer 3:20) and R Joseph B. Soloveitchik (Nefesh HaRav, p. 265).

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^[1]_{In general it is a Machloket Rishonim if Rabbenu Gershon ever prohibited polygamy in the case of Yibbum. R Yosef Karo rules that there is no prohibition, while R Moshe Isserlis rules that there still is a prohibition and Chalitza would be necessary (ShA EH 1:10). In our case where Chalitza is impossible, the claim is even R Isserlis would agree that it is permitted.}