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• Variable relation between rest energy density and string tension density is proposed. • The universe is found to be expanding and accelerating. • It is observed that isotropy of the universe is significantly affected by string. In this work we have proposed a variable relation between densities of cosmic string. Bianchi type II,VIII and IX space time is explored in the context of general relativity. The field equations are solved by assuming that the shear scalar is proportional to expansion scalar. Three different cases are investigated. It is observed that strings contribute significantly in isotropy and acceleration of the universe. [ABSTRACT FROM AUTHOR]