  We study the action of conformal transformations of the ambient space on the
Dirac operator coming into the Weierstrass (or spinor) representation of a
torus in the Euclidean four-space. It is showed that such an action generates a
flow acting on the potential of the operator, that this flow is described by a
nonlinear system of the Melnikov type and that it preserves the Floquet
multipliers of the Dirac operator with double-periodic potential. However this
flow is only almost isospectral since it does not preserve the spectral curve
in general and its action may result in adding or removing multiple points
corresponding to the same multipliers. We demonstrate that in some important
geometrical examples after reparameterization of the temporary variable such
flows are governed by integrable systems on whiskered tori.
