  We develop new constructions of 2D classical and quantum superintegrable
Hamiltonians allowing separation of variables in Cartesian coordinates. In
classical mechanics we start from two functions on a one-dimensional phase
space, a natural Hamiltonian $H$ and a polynomial of order $N$ in the momentum
$p.$ We assume that their Poisson commutator $\{H,K\}$ vanishes, is a constant,
a constant times $H$, or a constant times $K$. In the quantum case $H$ and $K$
are operators and their Lie commutator has one of the above properties. We use
two copies of such $(H,K)$ pairs to generate two-dimensional superintegrable
systems in the Euclidean space $E_2$, allowing the separation of variables in
Cartesian coordinates. All known separable superintegrable systems in $E_2$ can
be obtained in this manner and we obtain new ones for $N=4.$
