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Siegel introduced and studied the class of E-functions in 1929. They are power series, solutions of some linear differential equations, whose Taylor coefficients satisfy certain arithmetic and growth conditions. After the work of Siegel and Shidlovskii, and its refinement by Beukers, on the algebraic relations between values of E-functions over Q, it is important to know when the E-functions solutions of a given differential system of order 1 are algebraically dependent or not over Q(z). In this paper, we give the complete classification of the vector solutions of two-dimensional differential systems of order 1 whose components are algebraically dependent E-functions over Q(z). [ABSTRACT FROM AUTHOR]