  We implement the rotationally-invariant formulation of the two-dimensional
Hubbard model, with nearest-neighbors hopping $t$, which allows for the
analytical study of the system in the low-energy limit. Both U(1) and SU(2)
gauge transformations are used to factorize the charge and spin contributions
to the original electron operator in terms of the corresponding gauge fields.
The Hubbard Coulomb energy $U$ term is then expressed in terms of quantum phase
variables conjugate to the local charge and variable spin-quantization axis,
providing a useful representation of strongly correlated systems. It is shown
that these gauge fields play a similar role as phonons in the BCS theory: they
act as the "glue" for fermion pairing. By tracing out gauge degrees of freedom,
the form of paired states is established and the strength of the pairing
potential is determined. It is found that the attractive pairing potential in
the effective low-energy fermionic action is non-zero in a rather narrow range
of $U/t$.
