  One of the authors has recently introduced the concept of conjugate
Hamiltonian systems: the solution of the equation $h=H(p,q,t),$ where $H$ is a
given Hamiltonian containing $t$ explicitly, yields the function $t=T(p,q,h)$,
which defines a new Hamiltonian system with Hamiltonian $T$ and independent
variable $h.$ By employing this construction and by using the fact that the
classical Painlev\'e equations are Hamiltonian systems, it is straightforward
to associate with each Painlev\'e equation two new integrable ODEs. Here, we
investigate the conjugate Painlev\'e II equations. In particular, for these
novel integrable ODEs, we present a Lax pair formulation, as well as a class of
implicit solutions. We also construct conjugate equations associated with
Painlev\'e I and Painlev\'e IV equations.
