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Superintegrable classical Hamiltonian systems in two-dimensional Euclideanspace $E_2$ are explored. The study is restricted to Hamiltonians allowingseparation of variables $V(x,y)=V_1(x)+V_2(y)$ in Cartesian coordinates. Inparticular, the Hamiltonian $\mathcal H$ admits a polynomial integral of order$N>2$. Only doubly exotic potentials are considered. These are potentials wherenone of their separated parts obey any linear ordinary differential equation.An improved procedure to calculate these higher-order superintegrable systemsis described in detail. The two basic building blocks of the formalism arenon-linear compatibility conditions and the algebra of the integrals of motion.The case $N=5$, where doubly exotic confining potentials appear for the firsttime, is completely solved to illustrate the present approach. The general case$N>2$ and a formulation of inverse problem in superintegrability are brieflydiscussed as well.

Doubly Exotic $N$th-order Order Superintegrable Classical Systems Separating in Cartesian Coordinates