  We extend the recent work of Oliveras arXiv:2008.00940 and Oliveras &
Calatola-Young arXiv:2105.07580 to develop a new nonlocal formulation of the
water-wave problem for a three-dimensional fluid with a two-dimensional free
surface for an inviscid and irrotational fluid over a flat bathymetry. Using
this new formulation, we show how one can systematically derive Benjamin &
Olver's twelve conservation laws without explicitly relying on the underlying
Lie symmetries. This allows us to make draw new conclusions about conservation
laws and posit the potential existence of additional, nonlocal, conservation
laws for the water-wave problem.
