OpenChess

They are from YACPDB:
2RKQbk1/2QB1k1K/KQpk1BPN/RKpp1QPk/Bpkp1KQn/BkNRKBBN/k1BQP1K1/BBK1kN1Q w - -
K1Rr1B1k/n1KPP1kp/NkpK4/RNPBpkPK/RK1PkNBp/PB1k1K1R/PPkNPpKR/k1B1KBNr w - -

What do you mean by 'process'? It would be no problem to load these positions into Fairy-Max. But the only processing it will do is telling you that white is checkmated (in the first position). Which makes sense: black can capture a white King (i.e. white is in check), and no move can create a position where black cannot capture a white King. So according to FIDE rules, this means white is checkmated.

If that is not what you want, you are apparently not playing by FIDE rules. In that case you should define the rules under which you want to play this. Note that Fairy-Max can be configured to use extinction-Chess rules on multiple Kings, rather than absolute royalty, or even the 'duple-check' rule of Spartan Chess (where you are not allowed to leave more than one King in check if you have multiple Kings). But this is not the standard setting in normal Chess.

Why should a program accept an illegal position? The game of chess is defined as each side having 8 pawns, 2 knights, 2 bishops, 2 rooks, 1 queen and 1 king. The pawns can promote to anything bug a king. That means up to 10 knights, bishops or rooks, but no more than 14 of those three in total counting pawns...

Position 1 simply has too many pieces, too many kings, too many errors. Crafty rejects it as it should, since Crafty only plays real chess...

Position 2 is the same.

But there do exist engines that are not so limited...

You cannot blame a guy for asking which those are. The problem is just that the positions are not interesting.