From Tank Encyclopedia:
Mixed armament (gun and machine-gun) model 31
The first 10 pre-series vehicles were equipped with mixed armament, one of the turrets being equipped with a Russian variant PS1 or PS2 of the French Hotchkiss 37 mm (1.46 in) gun, designed by engineer P. Syachentov.
Other sources mention the PS-2 as being an "improved" model.
PC-2 could be a transliteration or transcription error: "C" in Russian is the letter "S" in English. Of course, the letter "P" in English looks a bit like the Greek "Pi" in Russian, so transliteration that correctly, but messing up the "C" is odd.
According to Wikipedia:
1-K was also a base for the 37 mm tank gun B-3 (5-K), the main armament of the BT-2 light tank.
Among other pieces, Rheinmetall brought to the USSR 12 37 mm anti-tank guns, which can be seen as an early variant of the PaK 35/36 – the most numerous anti-tank gun of the Wehrmacht until 1942. In the USSR the gun was designated 37 mm anti-tank gun model 1930 (1-K)
Hotchkiss is probably completely different from the Puteaux:
The Hotchkiss 37 mm/5 (5 = five-barrel) gun was first tested in Russia in 1879. Starting in 1884, the Russian Navy bought over 150 of these and the similar 47 mm five barrel gun for use as anti-torpedo boat weapons on large ships and to arm light craft. By 1884 there were 126 of the 37 mm/5 guns in service. Russian production of these guns was started at Tula in 1886 and by the time production stopped in 1896 an additional 290 weapons had been made. During World War I some of the surviving guns were used in the AA role.
These Hotchkiss weapons were crank-operated, five-barrel revolver-style rotating guns, externally somewhat similar to a Gatling gun but having a different ammunition feed system.
In addition to the five-barrel weapons, the Russians also purchased 37 mm/1 (1 = single barrel) Hotchkiss guns in 1883. By 1901 some 276 guns of this type were in service. As these single barrel guns were easier to manufacture than the multi-barrel guns, they became standard equipment in the Russian Navy until 1905. In that year, combat during the Russo-Japanese War proved these weapons to be ineffective and they were quickly removed from most larger ships. During World War I and the Russian Civil War they were widely used to arm light ships and river craft. Several of these weapons survived to serve in World War II on the ships of the Pinsk River flotilla.
These guns had monobloc barrels with a vertical blade type breech mechanism.
That's from NavWeaps.