  Superconductivity in strongly correlated electrons can emerge out from a
normal state that is beyond the Landau's Fermi liquid paradigm, often dubbed as
"non-Fermi liquid". While the theory for non-Fermi liquid is still not yet
conclusive, a recent study on the exactly-solvable Hatsugai-Kohmoto (HK) model
has suggested a non-Fermi liquid ground state whose Green's function resembles
the Yang-Rice-Zhang ansatz for cuprates [P. W. Phillips, L. Yeo and E. W.
Huang, Nat. Phys. $\bf{16}$, 1175 (2020)]. Similar to the effect of on-site
Coulomb repulsion in the Hubbard model, the repulsive interaction in the HK
model divides the momentum space into three parts: empty, single-occupied and
double-occupied regions, that are separated from each other by two distinct
Fermi surfaces. In the presence of an additional Bardeen-Cooper-Schrieffer
(BCS)-type pairing interaction of a moderate strength, we show that the system
exhibits a "two-stage superconductivity" feature as temperature decreases: a
first-order superconducting transition occurs at a temperature $T_{\rm c}$ that
is followed by a sudden increase of the superconducting order parameter at a
lower temperature $T_{\rm c}^{\prime}<T_{\rm c}$. At the first stage, $T_{\rm
c}^{\prime}<T<T_{\rm c}$, the pairing function arises and the entropy is
released only in the vicinity of the two Fermi surfaces; while at the second
stage, $T<T_{\rm c}^{\prime}$, the pairing function becomes significant and the
entropy is further released in deep (single-occupied) region in the Fermi sea.
The phase transitions are analyzed within the Ginzburg-Landau theory. Our work
sheds new light on unconventional superconductivity in strongly correlated
electrons.
