  We consider theoretically the nonlinear quantized Thouless pumping of a
Bose-Einstein condensate loaded in a two-dimensional dynamical optical
lattices. We encountered three different scenarios of the pumping: quasi-linear
one occurring for gradually dispersing wave packets, transport carried by a
single two-dimensional soliton, and multi-soliton regime when initial wave
packet splits into several solitons. The scenario to be realized depends on the
number of atoms in the initial wave packet and on the strength of the two-body
interactions. The magnitude and direction of the displacement of a wavepacket
are determined by Chern numbers of the populated energy bands and by the
inter-band transitions induced by two-body interactions. As a case example we
explore a separable potential created by optical lattices whose constitutive
sublattices undergo relative motion in the orthogonal directions. For such
potentials, obeying parity-time symmetry, fractional Chern numbers, computed
over half period of the evolution, acquire relevance. We focus mainly on
solitonic scenarios, showing that one-soliton pumping occurs at relatively
small as well as at sufficiently large amplitudes of the initial wavepacket,
while at intermediate amplitudes the transport is multi-solitonic. We also
describe peculiarities of the pumping characterized by two different
commensurate periods of the modulations of the lattices in the orthogonal
directions.
