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We consider boundary value problems for elliptic systems in the sense of Agmon-Douglis-Nirenberg on plane domains with corners, where the domain, the coefficients of the operators and the right hand sides all depend on a parameter. We construct corner singularities in such a way that the corresponding decomposition of the solution into regular and singular parts is stable, i. e. the regular part and the coefficient of the singular functions depend smoothly on the parameter. The construction of these singular functions continues the paper [3] and generalizes results known for second order scalar boundary value problems - see [4,5] [12].