Virasoro symmetries of a coupled rational reduced 2D Toda hierarchy

We introduce an interpolated coupled 2D Toda hierarchy and its symmetry reductions characterized by a rational factorization of the Lax operator. The reduced hierarchy contains several new coupled infinite-dimensional integrable hierarchies, such as the coupled bigraded Toda hierarchy, the coupled Ablowitz–Ladik hierarchy, and so on. After that, we construct additional symmetries of the coupled rational reduced 2D Toda hierarchy. These symmetries constitute a coupled Virasoro algebra and a Block Lie algebra.