Wolfes model aka $G_2\mathrm{/}{I}_6$-rational integrable model: $g^{(2)},g^{(3)}$ hidden algebras and quartic polynomial algebra of integrals

One-dimensional 3-body Wolfes model with 2- and 3-body interactions also known as $G_2\mathrm{/}{I}_6$-rational integrable model of the Hamiltonian reduction is exactly-solvable and superintegrable. Its Hamiltonian $H$ and two integrals ${\mathcal{I}}_1,{\mathcal{I}}_2$, which can be written as algebraic differential operators in two variables (with polynomial coefficients) of the 2nd and 6th orders, respectively, are represented as non-linear combinations of $g^{(2)}$ or $g^{(3)}$ (hidden) algebra generators in a minimal manner. By using a specially designed MAPLE-18 code to deal with algebraic operators it is found that $(H,{\mathcal{I}}_1,{\mathcal{I}}_2,{\mathcal{I}}_{12}\equiv [{\mathcal{I}}_1,{\mathcal{I}}_2])$ are the four generating elements of the {\it quartic} polynomial algebra of integrals. This algebra is embedded into the universal enveloping algebra $g^{(3)}$. In turn, 3-body/$A_2$-rational Calogero model is characterized by cubic polynomial algebra of integrals, it is mentioned briefly.