R. Fuchs' problem of the Painleve equations from the first to the fifth
  We will classify all rational transformations which change the confluent hypergeometric equations to linear equations of the Painleve type from the first to the fifth. We show such rational transformations correspond to almost all of algebraic solutions of the Painleve equations from the first to fifth up to the Backlund transformations. 
Yousuke Ohyama, Shoji Okumura