Commuting differential operators of rank two

Commuting differential operators with symbols ∂ 1 2 + ∂ 2 2 , ∂ 1 2 ∂ 2 2 are related to the functional equation V 1 ( x 1 ) U + ( x 1 + x 2 ) + V 1 ( x 1 ) U − ( x 1 − x 2 ) + V 2 ( x 2 ) U + ( x 1 + x 2 ) − V 2 ( x 2 ) U − ( x 1 − x 2 ) = F + ( x 1 + x 2 ) + F − ( x 1 − x 2 ) + G 1 ( x 1 ) + G 2 ( x 2 ). We discuss several properties, such as a meromorphic extension of solutions and periodicity, of this functional equation and commuting differential operators.