  A multi-Poisson structure on a Lie algebra $\mathfrak{g}$ provides a
systematic way to construct completely integrable Hamiltonian systems on
$\mathfrak{g}$ expressed in Lax form $\partial X_\lambda /\partial t =
[X_\lambda , A_\lambda ]$ in the sense of the isospectral deformation, where
$X_\lambda , A_\lambda \in \mathfrak{g}$ depend rationally on the indeterminate
$\lambda $ called the spectral parameter. In this paper, a method for modifying
the isospectral deformation equation to the Lax equation $\partial X_\lambda
/\partial t = [X_\lambda , A_\lambda ] + \partial A_\lambda /\partial \lambda $
in the sense of the isomonodromic deformation, which exhibits the Painlev\'e
property, is proposed. This method gives a few new Painlev\'e systems of
dimension four.
