When Knowledge of a Single Integral of Motion is Sufficient for Integration of Newton Equations $\ddot{q} = M (q)$

For an autonomous dynamical system of $n$ differential equations each integral of motion allows for reduction of the order of equations by 1 and knowledge of $(n - 1)$ integrals is necessary for the system to be integrated by quadratures. The amenability of Hamiltonian systems to being integrated by quadratures is characterised by the Liouville theorem where in $2n$-dimensional phase space only $n$ integrals are sufficient as equations are generated by 1 function — the Hamiltonian.
There are, however, large families of Newton-type differential equations for which knowledge of 2 or 1 integral is sufficient for recovering separability and integration by quadratures. The purpose of this paper is to discuss a tradeoff between the number of integrals and the special structure of autonomous, velocity-independent 2nd order Newton equations $\ddot{\boldsymbol{q}}=\boldsymbol{M}(\boldsymbol{q})$, $\boldsymbol{q}\in\mathbb R^n$, which allows for integration by quadratures.
In particular, we review little-known results on quasipotential and triangular Newton equations to explain how it is possible that 2 or 1 integral is sufficient. The theory of these Newton equations provides new types of separation webs consisting of quadratic (but not orthogonal) surfaces.