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creature

Eikþyrnir

The hart that stands by Valhalla browsing the tree Læraðr; from his antlers drips the water that fills Hvergelmir, source of all rivers.

Eikþyrnir is a stag who stands beside Valhalla with the goat Heiðrún. Both feed on the tree Læraðr. Heiðrún gives mead; Eikþyrnir gives water. Grímnismál says that "From his horns a stream into Hvergelmir drops, / Thence all the rivers run" Grímnismál 26.

Hvergelmir is the great spring that also lies under a root of Yggdrasil, so the stag's antlers stand at the head of the world's whole water system. Grímnismál follows the stanza with a long list of river names Grímnismál 27-28, and Snorri, retelling it, gives a list too: "Síd, Víd, Søkin, Eikin, Svöl, Gunnthrá" and many more, some flowing "about the abodes of the Æsir" Gylfaginning 39. Snorri says Eikþyrnir "stands in Valhall" rather than beside it.

That is all the sources say. Eikþyrnir is separate from the four harts that feed in Yggdrasil's branches, though the image is similar.

In the sources

Quoted from the hosted translations; each line is checked against the text when the site is built. Select a reference to read it in context.

Eikthyrnir is the hart who stands by Heerfather's hall And the branches of Lærath he bites; From his horns a stream into Hvergelmir drops, Thence all the rivers run.
Grímnismál 26 The hart whose antlers are the source of the world's rivers.
Even more worthy of note is the hart Eikthyrni, which stands in Valhall and bites from the limbs of the tree; and from his horns distils such abundant exudation that it comes down into Hvergelmir
Gylfaginning 39 Snorri's version, followed by a long list of river names.

Related

Heiðrún The goat that stands by Valhalla and browses the tree Læraðr; from her udders flows mead enough to make all the einherjar drunk every day. Hvergelmir The spring from which all rivers flow, fed by the dripping horns of a hart in Valhall; Snorri puts it in Niflheim under a root of Yggdrasil, swarming with serpents. Valhalla Valhöll Odin's vast hall of the battle-dead, roofed with shields and raftered with spears, where the einherjar feast and fight each day until they march out to meet the wolf at Ragnarök. The four stags of Yggdrasil Dáinn, Dvalinn, Duneyrr, Duraþrór Four harts that browse the topmost branches of Yggdrasil; two of them share names with dwarves. Yggdrasil Yggdrasill The world-ash at the centre of the Norse cosmos: its roots reach the realms of gods, giants and the dead, and the gods hold court beneath it every day.