We show that a new integrable two-component system of KdV type studied by Karasu (Kalkanli) et al. (arXiv: nlin.SI/0203036) is bihamiltonian, and its recursion operator, which has a highly unusual structure of nonlocal terms, can be written as a ratio of two compatible Hamiltonian operators. Using this, we prove that the system in question possesses an infinite hierarchy of local commuting generalized symmetries and conserved quantities in involution, and the evolution systems corresponding to these symmetries are bihamiltonian as well.