I found the original Mexican song by searching for "Ticuri cui, cui, cu." It's a choral piece, arranged by Victoria Ebel-Sabo, described as "really a love song, with the singer's heart tired from longing as the owl is tired from flying."

TECOLOTE

Tecolote de Guadaña
pájaro madrugador.
Si tuviera tus alitas
pa´ ir a ver a mi amor.
Ticuri cui, cui, cu.
Pobrecito tecolote,
ya se cansa de volar.


Tecolote of Guadaña,
bird of the dawn.
If I only had your wings
to go and see my love.
Coo coo.
Poor little owl,
you are tired from flying.


Source: http://www.harmonium.org/prognote/200603.pdf

---

This song was recorded by Cynthia Gooding as "El Tecolote." It's on her Elektra album titled Sings Spanish, Mexican, and Turkish Folk Songs. The lyrics are far more complicated, and I have no hope of transcribing it.

HOWEVER, you will find several versions of songs titled "El Tecolote" in a book titled Estudio y clasificación de la música tradicional Hispánica de Nuevo México By Vicente T. Mendoza, Virginia R. R. de Mendoza.

You will also find a number of versions of this song in a Mudcat thread titled Baby Owlet. Mudcatter "Q" did some terrific research on this song, and it's all in the "Owlet" thread.

There's a recording titled Tecolote de Guadaña by Frank McCulloch y sus Amigos. You'll find it on Spotify, and it's a kick to listen to.