  A prototypical example of a rogue wave structure in a two-dimensional model
is presented in the context of the Davey-Stewartson~II (DS~II) equation arising
in water waves. The analytical methodology involves a Taylor expansion of an
eigenfunctionof the model's Lax pair which is used to form a hierarchy of
infinitely many new eigenfunctions. These are used for the construction of
two-dimensional (2D) rogue waves (RWs) of the DS~II equation by the even-fold
Darboux transformation (DT). The obtained 2D RWs, which are localized in both
space and time, can be viewed as a 2D analogue of the Peregrine soliton and are
thus natural candidates to describe oceanic RW phenomena,as well as ones in 2D
fluid systems and water tanks.
