“Do Not Lament Me, O Mother” is appointed to be sung at the Divine Liturgy on Great and Holy Saturday (the same text appears in the Canon at Matins of the Lamentations). I haven't been able to unearth any information on this particular setting or its composers, but it's my impression that it’s a contemporary composition. You can hear a church choir in Russia singing it in Slavonic in this YouTube video.
This is a composition of mine from around 2006 or so. It is a SSA trio of the exaposteilarion of Great and Holy Friday, in the Slavonic style, and using a translation from the Slavonic rather than the Greek (the Slavonic version interposes the word “wise,” for example). The hymn is appointed to be sung three times. Since the 19th century, it has been a common Russian practice that at least one of these repetitions should be done as a trio, hence the present version.
I’ve never heard of any parish actually performing this piece liturgically.
The default setting of the Exaposteilarion of Great and Holy Friday for those in the Russian tradition seems to be one of the Kievan chant settings, so why am I adding my voice to the din? Well, for years I have thought about composing a full-choir companion setting to my own treble-trio version. Then a couple weeks ago I realized that in fact the Kievan chant setting would work just fine with it.