  Symplectic integrators are widely implemented numerical integrators for
Hamiltonian mechanics, which preserve the Hamiltonian structure (symplecticity)
of the system. Although the symplectic integrator does not conserve the energy
of the system, it is well known that there exists a conserving modified
Hamiltonian, called the shadow Hamiltonian. For the Nambu mechanics, which is a
kind of generalized Hamiltonian mechanics, we can also construct
structure-preserving integrators by the same procedure used to construct the
symplectic integrators. In the structure-preserving integrator, however, the
existence of shadow Hamiltonians is nontrivial. This is because the Nambu
mechanics is driven by multiple Hamiltonians and it is nontrivial whether the
time evolution by the integrator can be cast into the Nambu mechanical time
evolution driven by multiple shadow Hamiltonians. In this paper we present a
general procedure to calculate the shadow Hamiltonians of structure-preserving
integrators for Nambu mechanics, and give an example where the shadow
Hamiltonians exist. This is the first attempt to determine the concrete forms
of the shadow Hamiltonians for a Nambu mechanical system. We show that the
fundamental identity, which corresponds to the Jacobi identity in Hamiltonian
mechanics, plays an important role in calculating the shadow Hamiltonians using
the Baker-Campbell-Hausdorff formula. It turns out that the resulting shadow
Hamiltonians have indefinite forms depending on how the fundamental identities
are used. This is not a technical artifact, because the exact shadow
Hamiltonians obtained independently have the same indefiniteness.
