  We derive formulas for the construction of all inequivalent Jacobian elliptic
fibrations on the Kummer surface of two non-isogeneous elliptic curves from
extremal rational elliptic surfaces by rational base transformations and
quadratic twists. We then show that each such decomposition yields a
description of the Picard-Fuchs system satisfied by the periods of the
holomorphic two-form as either a tensor product of two Gauss' hypergeometric
differential equations, an Appell hypergeometric system, or a GKZ differential
system. As the answer must be independent of the fibration used, identities
relating differential systems are obtained. They include a new identity
relating Appell's hypergeometric system to a product of two Gauss'
hypergeometric differential equations by a cubic transformation.
