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We consider a hyperk\"{a}hler $8$-manifold admitting either a $U(1) \times\mathbb{R}$, or a $U(1) \times U(1)$ action, where the first factor preserves$g$ and $I$, and acts on $\omega_2+i\omega_3$ by multiplying it by itself,while the second factor preserves $g$ and acts triholomorphically. Such datacan be reduced to a single function $H$ of two complex variables and two realvariables satisfying $6$ equations of Monge-Ampere type, which can be compactlywritten down using a Poisson bracket.

An Ansatz for Hyperk\"ahler $8$-Manifolds with two Commuting Rotating Killing Fields