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We consider strings living in $AdS_3\times S^3\times S^3\times S^1$ withnonzero $B$-field. By using specific ansatz for the string embedding, we obtaina class of solutions corresponding to strings moving in the whole tendimensional space-time. For the $AdS_3$ subspace, these solutions are given interms of incomplete elliptic integrals. For the two three-spheres, they areexpressed in terms of Lauricella hypergeometric functions of many variables.The conserved charges, i.e. the string energy, spin and angular momenta, arealso found.

String solutions in $AdS_3\times S^3\times S^3\times S^1$ with $B$-field