  We describe a family of generalized almost structures associated with a
Monge-Amp\`ere equation for a stream function of 2D incompressible fluid flows.
Using an indefinite metric field constructed from a pair of $2$-forms related
to the Monge-Amp\`ere equation, we show the existence of generalized
metric-compatible structures in our family of generalized structures. The
integrability of isotropic structures on the level of Dirac structures and
differential forms is discussed.
