  The BCS theory models electron correlations with pure zero-momentum pairs.
Here we consider a family of pairing Hamiltonians, where the electron
correlations are modelled with pure arbitrary-momentum pairs. We find all
models in the family are exactly solvable, and present these solutions. It is
interesting to note that the $\eta$ pair or the $d$ -wave pair condensate in
$T_{c}$ superconductivity can be the ground state of a Hamiltonian in the
family. These models are two-dimensional because only the z-component of the
total electron spin $S_z$ is conserved. Significantly, for the $\eta$ pair or
$d$ -wave pairing model in the family we find an analytical expression of
energy and an abrupt ground state change from independent particle state to the
$d$ -wave pair condensate, suggesting a quantum phase transition.
