Local bi-integrability of bi-Hamiltonian systems, Part II: Real smooth case

We prove that any bi-Hamiltonian system $v = \left(\mathcal{A} + \lambda \mathcal{B}\right)dH_{\lambda}$ on a real smooth manifold that is Hamiltonian with respect all Poisson brackets $\left(\mathcal{A} + \lambda \mathcal{B}\right)$ is locally bi-integrable. We construct a complete set of functions $\mathcal{G}$ in bi-involution by extending the set of standard integrals $\mathcal{F}$ consisting of Casimir functions of Poisson brackets, eigenvalues of the Poisson pencil, and the Hamiltonians. Moreover, we show that at a generic point of $M$ differentials of the extended family $d \mathcal{G}$ can realize any bi-Lagrangian subspace $L$ containing the differentials of the standard integrals $d \mathcal{F}$.