Half the dialogue was ad-libbed.
Etsilopp, the weapon-wielding class of aliens that can lord over lowly classes, is backward for the Russian word for police.
Wef's cap is actually the codpiece from the discarded fighter pilot's suit, and Pepelats was just thrown together from scrap iron from the nearest dump welded together by a local plumber.
The movie has been released on DVD in Russia but has gained virtually no notice elsewhere, with the exception of Japan. This is largely due to the fact that there was no official release of the movie with English subtitles for a long time. While the movie was in the making, a censorship threat emerged due to the use of the word Ku (Koo) which sounded like the initials "K. U." of the General Secretary of the Communist Party of the USSR at the time, K. U. Chernenko. Chernenko who assumed the leadership in 1984, died in 1985, and this removed the danger.
Originally Fiddler was supposed to give the music professor presents from his trip in Abkhazia along with the violin that he forgot because he got drunk. One of the presents was Chacha, Georgian brandy. Right when the film was going to be released Mikhail Gorbachev started his anti-alcohol campaign and they had to change Chacha to vinegar.