Bi‐Hamiltonian formulation for the Korteweg–de Vries hierarchy with sources

The first Hamiltonian structure is constructed for the Korteweg–de Vries (KdV) hierarchy with sources and for the respective modified systems. Applying the appropriate Miura map the second Hamiltonian structure for the KdV hierarchy with sources is derived. Stationary projection of considered system gives so‐called restricted flows of the KdV hierarchy as well as its bi‐Hamiltonian structure.