Gúthwinë

From the One Wiki to Rule Them All, the Lord of the Rings Encyclopedia.

Gúthwinë (Old English battle-friend) was a sword that belonged to Éomer. It was borne by him at the battle of Hornburg, and presumably at the Battle of the Pelennor Fields and the battle before Morannon.
In The Two Towers (film), Gúthwinë is shown as having a fullered blade and both the guard and the pommel were of bronze. The guard was shaped like horses' heads, one on each side of the blade.

Note- In The Two Towers (film), if you watch closely, you can see Gúthwinë slip out of its sheath as Éomer remounts after talking to Aragorn, Legolas, and Gimli.