Was Levirate Marriage Only For Single Men?

Levirate marriage was a biblical custom described in Deuteronomy 25:5–10:

5: If brothers dwell together, and one of them dies and has no son, the widow of the dead man shall not be married to a stranger outside the family; her husband's brother shall go in to her, take her as his wife, and perform the duty of a husband's brother to her.

6: And it shall be that the firstborn son which she bears will succeed to the name of his dead brother, that his name may not be blotted out of Israel.

7: But if the man does not want to take his brother's wife, then let his brother's wife go up to the gate to the elders, and say, 'My husband's brother refuses to raise up a name to his brother in Israel; he will not perform the duty of my husband's brother.'

8: Then the elders of his city shall call him and speak to him. But if he stands firm and says, "I do not want to take her,"

9: then his brother's wife shall come to him in the presence of the elders, remove his sandal from his foot, spit in his face, and answer and say, "So shall it be done to the man who will not build up his brother's house."

10: And his name shall be called in Israel, "The house of him who had his sandal removed." (NKJV).

If a married man in Old Testament Israel died, leaving behind a childless widow, the brother of the deceased was required by biblical law to marry his dead brother's widow. The purpose was to hopefully produce offspring that would become heirs to the wealth of the deceased.

The question I wish to focus on is whether or not levirate marriage was limited to single men. In other words, if the brother of the deceased was already married, was he still required to take his brother's widow as a second wife? Was it even allowed?

People who hold the position that polygamy was, is, and always will be a sin obviously claim these are mute questions because if polygamy itself is a sin, then taking more than one wife, even under the set of circumstances prescribed by levirate law, would also be sinful. Since the Bible itself is silent on this matter, unfortunately, the best we can do is to make an educated speculation. Nevertheless, if we can make a strong case that being married did not exempt a man from levirate marriage, then it should put to rest, once and for all, the idea that polygamy is always a sin.

What Do Commentaries Say About Married Men And Levirate Marriage?
Most commentaries say nothing about whether or not a married man was required to take the widow of his deceased brother as a secondary wife. But here's what I've found so far.

According to The Reformation Study Bible:

The limitation to brothers "living together" may indicate that it applied to an unmarried brother, but it is doubtful that this limitation held in practice.

According to The MacArthur Study Bible:

Obviously, this required that the brother be unmarried...

Here we have two different study Bibles taking opposite positions. The Reformation Study Bible says "it is doubtful that this limitation [an unmarried brother] held in practice," while The MacArthur Study Bible says "Obviously, this required that the brother be unmarried...".

According to the Oxford Dictionary of the Bible:

The [levirate] law did not forbid a man to be married twice (Deuteronomy 21:15-17 ) .