I put in Lampada's text into Babelfish and got this:

Stork with us lived summer, A it in winter stayed somewhere. Hippopotamus opened the mouth: Hippopotamus requests rolls. Sparrow requested crow to call wolf to the telephone. Fungus grows among the path, head on the thin leg. Woodpecker lived in the hollow empty, oak hollowed as by chisel. Fir tree to the hedgehog is similar: Hedgehog in the needles, fir tree - also. Beetle fell and cannot arise. It awaits, who will help it. Stars we saw in the daytime across the river, above the Kremlin... Hoarfrost lay on the branch of fir tree, needle within the night they became white. Tomcat caught mice and rats rabbit sheet cabbage it gnawed. Boats on the sea will sail, people by oars row. Bear found honey in the scaffolding, it is small honey, there are many bees. Rhinoceros butts by horn. You do not joke with the rhinoceros! Oslik was today the ashes: It learned, that it settled. Tortoise bears armor, it hides head from the fear. Digs the earth gray mole - it ruins vegetable-garden. Will sleep quietly old elephant - costing to sleep it knows how it. Cockroach lives after the stove, how warm place! Student taught lessons - he has in the inks of cheek. Fleet will swim to the native earth. Flag aboard each ship. Walk on the scaffolding polecat, predatory small little beast. The heron, important, big-nosed, the whole day stands as statue. Watchmaker, after squinting eye, makes hours or so for us. Schoolboy, schoolboy, you is the strong person: The sphere of terrestrial you bear as ball! By brush virgin land of 4 puppies, To shchekochu to it side. This button and lace - electric bell. Young - future sailor - southern small fishes to us it brought. There are no berries acid than the cranberry. 4 to the memory I know letters.
So it seems like the rhymes are very subtly related to the themes of communism? Possibly, the translations of the rhymes will need an additional text explaining the reference. Personalities like Lenin, the proletariat, the capitalist, the White Russians, etc are not referred to directly but instead as childhood nursery rhyme characters like the stork, bear or rhinoceros. That or this translation is really terrible.



The fine print does say 1921 which is definitely Lenin-era. Also, the art depicts the royals, the clergy and the White generals as enemies of the people which definitely reflects that time period; Stalin-era works would presumably focus more on capitalism.