On differential equations invariant under a projective transformation group: integrability and reductions

We consider a projective transformation and establish the invariants for this transformation group up to order seven. We use the obtained invariants to construct a class of nonlinear evolution equations and identify some symmetry-integrable equations in this class. Notably, the only symmetry-integrable evolution equation of order three in this class is a fully-nonlinear equation for which we find the recursion operator and its connection to the Schwarzian KdV. We furthermore establish that higher-order symmetry-integrable equations in this class belong to the hierarchy of the fully-nonlinear 3rd-order equation and prove this for the 5th-order case as well as for the quasi-linear 7th-order case. We list all symmetry reductions of this 3rd-order fully-nonlinear symmetry-integrable evolution equation to ordinary differential equations by exploiting the 1-dimensional optimal Lie symmetry subalgebras of the transformation group. We also identify the ordinary differential equations that are invariant under this projective transformation and reduce the order of these equations. 33 pages. An extended version