We consider confluences of Euler-type integrals expressing solutions to Appell's F_2 system of hypergeometric differential equations, and study systems of confluent hypergeometric differential equations of rank four of two variables. Our consideration is based on a confluence transforming the abelian group (C×)~2 to the Jordan group of size two. For each system obtained by our study, we give its Pfaffian system with a connection matrix admitting a decomposition into four or five parts, each of which is the product of a matrix depending only on parameters and a rational 1-form in two variables. We classify these Pfaffian systems under an equivalence relation. Any system obtained by our study is equivalent to one of Humbert's Ψ_1 system, Humbert's Ξ_1 system, and the system satisfied by the product of two Kummer's confluent hypergeometric functions.
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