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We calculate the dynamical response of two-layer Josephson arrays to a biascurrent applied into only one layer (primary layer), in a perpendicularmagnetic field. The pancake vortices in the two layers sometimes form into fluxlines which move as a unit under the influence of the bias current. The linesbreak apart into independent pancakes in sufficiently high currents,sufficiently weak interlayer coupling, or possibly sufficiently high fields. Wediscuss the relevance of these results to recent experiments in high-T$_c$superconductors in the flux transformer geometry.Comment: 12 pages + 8 figures, REVTeX 3.0, submitted to Phys. Rev.

2D-3D vortex-line transition in two-layer Josephson arrays