  We study controllability issues for the 2D Euler and Navier-Stokes (NS)
systems under periodic boundary conditions. These systems describe motion of
homogeneous ideal or viscous incompressible fluid on a two-dimensional torus
$\mathbb{T}^2$. We assume the system to be controlled by a degenerate forcing
applied to fixed number of modes.
  In our previous work \cite{ASpb,AS43,ASDAN} we studied global controllability
by means of degenerate forcing for Navier-Stokes (NS) systems with nonvanishing
viscosity ($\nu >0$). Methods of differential geometric/Lie algebraic control
theory have been used for that study.
  In the present contribution we improve and extend the controllability results
in several aspects: 1) we obtain a stronger sufficient condition for
controllability of 2D NS system in an observed component and for
$L_2$-approximate controllability; 2) we prove that these criteria are valid
for the case of ideal incompressible fluid ($\nu=0)$; 3) we study solid
controllability in projection on any finite-dimensional subspace and establish
a sufficient criterion for such controllability.
