Rational implicit advection stencil with exact shift exceptions ($\mu =\Delta t/\Delta x$)

Consider a one-step stencil with new-level coefficients $a=\mu (1+\mu )/2$, $b=(1+\mu )(2-\mu )$ and $c=(1-\mu )(2-\mu )/2$ at offsets $-1,0,1$, and old-level coefficients $d=2-\mu$, $e=1+\mu$ at offsets $0,1$. For $u_t=u_x$, its exact-solution residual starts at fourth degree with coefficient $\mu (\mu -2)(\mu -1)(\mu +1)/24$. Thus its normalized local truncation error is third order at fixed nonzero Courant number, except at the exact-shift values $-1,1,2$; zero gives the zero-step identity. For its Fourier symbol, $|D{|}^2-|N{|}^2=\mu (\mu -2)(\mu -1)(\mu +1)(1-\cos \theta {)}^2$. Uniformly invertible stable steps occur for $\mu \le -1$, $0\le \mu \le 1$ or $\mu \ge 2$.