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In order for the quaternionic structure on a hyperkahler manifold to take the canonical form $$ J_1= \left[\begin{array}{cccc} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1\\ 0 & 0 & 1 & 0 \end{array}\right] $$ $$ J_2= \left[\begin{array}{cccc} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \end{array}\right] $$

$$ J_3= \left[\begin{array}{cccc} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0 \end{array}\right] $$

in a local patch, the overlap maps between coordinate charts ought to satisfy the Cauchy-Fueter equations, i.e., they ought to be affine.

Besides tori, what hyperkaehler manifolds admit an atlas with affine overlap maps, and therefore admit quaternionic structure with the canonical form written above?

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