  A hypersymplectic structure on a 4-manifold is a triple $\omega_1, \omega_2,
\omega_3$ of 2-forms for which every non-trivial linear combination
$a^1\omega_1 + a^2 \omega_2 + a^3 \omega_3$ is a symplectic form. Donaldson has
conjectured that when the underlying manifold is compact, any such structure is
isotopic in its cohomolgy class to a hyperk\"ahler triple. We prove this
conjecture for a hypersymplectic structure on $T^4$ which is invariant under
the standard $T^3$ action. The proof uses the hypersymplectic flow, a geometric
flow which attempts to deform a given hypersymplectic structure to a
hyperk\"ahler triple. We prove that on $T^4$, when starting from a
$T^3$-invariant hypersymplectic structure, the flow exists for all time and
converges modulo diffeomorphisms to the unique cohomologous hyperk\"ahler
structure.
