This is a repost
The following scene from Sienkiewicz's Knights of the Cross (which takes place in 1410) made me think whether such a situation was possible at all. What would Zawisza do with his horse? Would members of his lance remain close to him all the time or was it more likely that the lance became separated during combat?
Under the gigantic Arnold von Baden, who was surrounded by country infantry, a rampart of Polish bodies had been piled up, while he, mighty and invincible, stood above it, as stands a boundary pillar on an eminence, at last Zavisha Charny himself came to him; but seeing the knight without a horse, and not wishing to attack him from behind contrary to knightly usage, he sprang off his horse and called to him from a distance.
"Turn thy head, German, and surrender, or meet me."
Arnold turned and recognizing Zavisha by his black armor, and his shield, said in his gloomy soul,—
"Death is present, and my hour has come, for no one can escape that man alive. But if I could conquer him I should win immortal glory, and save my life perhaps."
Then he sprang toward him and they struggled like two tempests on that ground covered with corpses. But Zavisha surpassed all men in strength so tremendously that unfortunate were the parents to whose children it happened to meet him in battle.
I saw your earlier post and hesitated to comment, because I am emphatically not a scholar of medieval Poland.
I think it's a very fanciful and romantic situation, but one that would rarely, if ever, occur in the real world. Within the context of pitched battles, medieval soldiers, knights or otherwise, fought in formations, not as individuals. They kept place with the men to their left and right and followed the orders of their immediate leaders (the degree to which higher echelon commanders could exert their authority is a matter of dispute). Breaking off to engage the enemy in single combat is a good way to weaken your unit and get yourself killed.
Now, with that said, there certainly are examples of soldiers seeking each other on the battlefield. The most obvious would be pre-battle duels: I know the Scots and English are supposed to have sent champions to fight each other before the Battle of Bannockburn. But even after the battle had opened, individual men (and their units) might be targeted, if they were conspicuous and valuable enough. At Poitiers, the English men-at-arms surged forward against John II's bodyguard, killing Geoffroi de Charny (probably the most famous knight in France at the time) and capturing both the king and his standard, the oriflamme. Similarly, at Agincourt Jean, Duke of Alencon led an attack on Henry V's bodyguard. The sudden assault nearly succeeded, but the English held, and cut Jean down in a vengeful wrath.