  Type III multi-step rationally-extended harmonic oscillator and radial
harmonic oscillator potentials, characterized by a set of $k$ integers $m_1$,
$m_2$, \ldots, $m_k$, such that $m_1 < m_2 < \cdots < m_k$ with $m_i$ even
(resp.\ odd) for $i$ odd (resp.\ even), are considered. The state-adding and
state-deleting approaches to these potentials in a supersymmetric quantum
mechanical framework are combined to construct new ladder operators. The
eigenstates of the Hamiltonians are shown to separate into $m_k+1$
infinite-dimensional unitary irreducible representations of the corresponding
polynomial Heisenberg algebras. These ladder operators are then used to build a
higher-order integral of motion for seven new infinite families of
superintegrable two-dimensional systems separable in cartesian coordinates. The
finite-dimensional unitary irreducible representations of the polynomial
algebras of such systems are directly determined from the ladder operator
action on the constituent one-dimensional Hamiltonian eigenstates and provide
an algebraic derivation of the superintegrable systems whole spectrum including
the level total degeneracies.
