Lie group analysis of a generalized Krichever-Novikov differential-difference equation

The symmetry algebra of the differential–difference equation $${\dot{u}}_n=[P(u_n)u_{n+1}{u}_{n-1}+Q(u_n)(u_{n+1}+u_{n-1})+R(u_n)]\mathrm{/}(u_{n+1}-u_{n-1}),$$ where $P$, $Q$ and $R$ are arbitrary analytic functions is shown to have the dimension $1\le dimL\le 5$. When $P$, $Q$ and $R$ are specific second order polynomials in $u_n$ (depending on 6 constants) this is the integrable discretization of the Krichever–Novikov equation. We find 3 cases when the arbitrary functions are not polynomials and the symmetry algebra satisfies $dimL=2$. These cases are shown not to be integrable. The symmetry algebras are used to reduce the equations to purely difference ones. The symmetry group is also used to impose periodicity $u_{n+N}=u_n$ and thus to reduce the differential–difference equation to a system of $N$ coupled ordinary three points difference equations.