The authors analyze plane boundary value problems for equilibrium states of elastic mixtures. The complex vector forms of the basic equations and the boundary conditions are presented. Kolosov-Muskhelishvili type representations of displacements and stresses are used to derive the Fredholm-type integral equations of second kind for three basic two-dimensional boundary value problems (given boundary displacements; given boundary tractions; on one part of the boundary the displacements are known and on the other part of boundary the tractions are known). The solvability, existence and uniqueness problems of derived integral equations are discussed.