Integrability conjecture for the tabulated homogeneous Newton systems

Let $k$ and $\lambda$ be such that $(k,\lambda)$ belongs to an item of the paper's table. Consider the Newton equations

$$\dot q_1=p_1,\quad \dot p_1=-\lambda q_1q_2^{k-2},\qquad \dot q_2=p_2,\quad \dot p_2=q_2^{k-1}.$$

Integrability conjecture. These equations are integrable in the Jacobi sense with polynomial first integrals $I_1$ and $I_2$, where

$$I_1=\frac{1}{2}p_2^2+\frac{1}{k}q_2^k.$$

Moreover, for every $M>0$ there is a value of $\lambda$ for which the degree of $I_2$ in the momenta is greater than $M$, while no additional polynomial first integral independent of $I_1$ has degree in the momenta at most $M$. If $(k,\lambda)$ belongs to an item other than item 1 of the table, there are two additional polynomial first integrals $I_2$ and $I_3$ that are functionally independent together with $I_1$. The claim concerns selected parameter pairs in the table and gives evidence that the necessary integrability conditions found earlier may be sufficient for these families; the stated existence and degree assertions are not established here.