Need help identifying/restoring wood finish on a 1980s Soviet chess set

Anyone out there familiar with this era of Soviet chess sets, their construction and finish, or how they were most likely making chess sets in the USSR back then? General or chess-specific wood finish knowledge and experience also appreciated. I have this set which I want to refinish; I just tested it with acetone to determine what kind of finish it is, and got no discernible result. I tried isopropyl alcohol too, with seemingly no effect. I would just like to know what I'm working with so as not to damage them further, because I adore the design of the pieces and it took me weeks of searching to find this set.

I saw some woodworking & finishing talk on these forums, so here's hoping someone who knows something will find this...

More info:

I just got a chess a couple weeks ago. It was purchased from an Etsy seller in Ukraine and shipped to the US. The listing didn't have any details other than the typical SEO word-salad. When it arrived, seeing it firsthand, I found that the finish on the white pieces especially is quite cracked (worse than you can tell in these photos, actually downright nasty in the cold light of day). Not to mention the felt pads look like scum, and the finials on the kings were cracked off, replaced with poorly centered plastic beads, and (I think) actually swapped onto the Queens. So, I don't feel guilty about tampering with them more as a little project.

Anyway, here are some photos. I know there are several different versions of this style/era of set—I've seen some variations in the shape of the knights—and I don't even really know what year it is from.

There's a confusing dateline which says OCT 13-39-80 as well as a separate stamp overlapping which looks like it probably says 1983.

I remember reading about someone who was able to strip down an old Soviet set. I forgot what solution he used. I remember he said it was really toxic and had to do the work outside. I tried to find the thread but I couldn't find it. I found this other one. Perhaps you can DM the OP. See post 63 https://www.chess.com/forum/view/chess-equipment/where-can-i-get-soviet-style-chess-pieces-like-these?page=4.

Also, @Cgrau is an expert in all things Soviet chess sets. He could probably give you some background info about materials. Based upon what I read, @Greghunt is very knowledgeable about restoration work. There are a few others but I can't remember off the top of my head. Good luck!

I get where you're coming from, but if it were my set I'd leave it as it is and enjoy the patina of age and history. It's been played with and I love that about vintage and antique chess men.

Regarding the stamps and the dating... The square stamp actually relates specifically to the board, Board No. 3, which according to one dealer/expert means the board has 30mm large squares. You can measure yours to test this. I'd have to transliterate the stamp to tell you the factory and Oblast of origin. Even so, the stamp is some evidence of the date of manufacture of the set, inasmuch as it is not unreasonable to infer that the board and the pieces were purchased together, even though it remains possible that they were purchased separately. I don't have any information regarding that style of chessmen.

The finishes on Soviet sets are all over the place. I'm concerned that many of the black ones contain lead paint, and hesitate to work on them myself. I don't consider myself any kind of expert on them, and tend to refer my questions and work to artists like Alan Power of The Chess Schach, Porat Jacobson of Jacobson Handmade Chess Sets, and Alan Dewey of Chessspy.

The absolute finish on any of my Soviet sets appeared on what also is one of my most interesting, a very early Grandmaster 3 model one dealer/expert calls a "Supreme Grandmaster" set, I suppose in contrast to the so-called "Grandmaster Supreme" sets. In any event, the finish on both sides was very sticky to the touch. Very annoying to play with. I decided to gently wash off some of the accumulated grime with a slightly wet rag. I confess I was pissed so I took a black bishop and stripped it with a non-fuming stripping agent. It was a lot of work. I wound up sending the set to the aforementioned Alan Power in Toronto, who stripped all the black pieces and refinished them. He told me they had used a water-based varnish. The stuff was complete goo. It was like the black pieces were caked in black shoe polish. If I were you, I'd contact Alan Power and consult with him about what you should do.

I am a minimalist when it comes to these sets, and I value their natural state as homage to their makers and past owners. But I also play with many of my sets, and I will do what I must to get them into a playable state. On your set, what I would do personally is to get replacement finials made and leave the finish alone. Your mileage may vary. It's a personal choice.

Thanks so much @MCH818, @fixbayonets, and @cgrau for your responses, that is way more detailed help than I ever expected to get, and so soon. To your points, fixbayonets and cgrau, I tend to agree completely with leaving things as-is and would normally not at all mind something looking "old," but in this case I wonder if the finish actually cracked very recently due to dramatic temperature changes (in transit out of Ukraine and across the Atlantic by air in Nov/Dec). And a couple pieces simply look too nasty. One white pawn in particularly has been absolutely spattered with blotches of yellow, and they all somehow have weird dirty glue goo creeping up from the felt.

I had not thought about lead paint, so thanks again cgrau. I will definitely avoid messing with the black in particular.

Looking at these again in daylight, I just noticed a couple more things:

1. The acetone test I did yesterday actually did remove some of the finish, noticeable because the spot is much less yellow, but the cracking still there. Unsure if this means the wood is cracked (weird pattern if so) or if I'm looking at two layers of different finishes.

2. The orange tint of the finish is very noticeable where it collected thicker in the crevices of the black pieces. I would have guessed this means it's more likely shellac, but the acetone test didn't make it tacky; more like it dissolved with some hard scrubbing. And the alcohol test didn't dissolve anything, which it should've with shellac. 

tl;dr I think (the outermost layer, at least) is a tinted lacquer or some kind of lacquer mix.

In the process of writing this, I think I've decided not to risk making the the set smell like lacquer thinner and instead spend my days gently scraping off the scummier looking parts, maybe test stripping the rest of the acetone off this knight, replace the felt and finials myself. The cracks I can appreciate. I think it's the cracks + bad finish application that's bothering me.

Also @cgrau very interesting notes about the stamps, however the squares measure at closer to 33.33mm or 34.5mm (depending on if I'm measuring individual squares or the whole grid and dividing by eight for the average ...don't ask me why these are different).

I would like to translate the stamp, as I'm really wondering about that "39" in the date.

I find your set to be charming. There is something to be said for having a set with the original finish even if it is cracking or melting away.

The cracking is in the finish, not the wood. It's very common in these sets.

I like the minimalist plan you propose.