Joint complete monotonicity of rational functions in two variables and toral $m$-isometric pairs

We discuss the problem of classifying polynomials $p:{\mathbb{R}}_{+}^2\to (0,\mathrm{\infty })$ for which $\frac{1}{p}=\{\frac{1}{p(m,n)}{\}}_{m,n\ge 0}$ is joint completely monotone, where $p$ is a linear polynomial in $y\mathrm{.}$ We show that if $p(x,y)=a+bx+cy+dxy$ with $a>0$ and $b,c,d\ge 0,$ then $\frac{1}{p}$ is joint completely monotone if and only if $ad-bc\le 0.$ We also present an application to the Cauchy dual subnormality problem for toral $3$-isometric weighted $2$-shifts.