The Painlevé III3 and III1 Lax pairs reduce at solution singularities to the modified Mathieu and Doubly Confluent Heun equations, matching SU(2) N=2 SYM with Nf=0 and Nf=2 in the Nekrasov-Shatashvili background.
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From Painlev\'e equations to ${\cal N}=2$ susy gauge theories: prolegomena
The Painlevé III3 and III1 Lax pairs reduce at solution singularities to the modified Mathieu and Doubly Confluent Heun equations, matching SU(2) N=2 SYM with Nf=0 and Nf=2 in the Nekrasov-Shatashvili background.