Quadratic geodesic first integral ($K(x,p)=K^{ij}(x)p_i{p}_j,\{K,H\}=0$)

A quadratic homogeneous polynomial in the fibre variables of the cotangent bundle is a first integral of the geodesic Hamiltonian exactly when its symmetric lowered coefficients form a rank-two Killing tensor. For $v=g^{-1}p$, the Poisson bracket is $\{K,H\}={\nabla }_{(a}{K}_{bc)}{v}^a{v}^b{v}^c$, so the equivalence follows by comparing cubic coefficients at every point.