  The possibility of realizing pair density wave (PDW) phases, in which Cooper
pairs have a finite momentum, presents an interesting challenge that has been
studied in a wide variety of systems. In conventional superconductors, this is
only possible when external fields lift the spin degeneracy of the Fermi
surface, leading to pair formation at an incommensurate momentum. Here, we
study a second possibility, potentially relevant to transition metal
dichalcogenides, in which the Fermi surface consists of a pair of pockets
centered at the $\pm K$ points of the Brillouin zone as well as a central
pocket at the $\Gamma$ point. In the limit where these three pockets are
identical, the pairing susceptibility has a logarithmic divergence at the
non-zero wave-vectors $\pm \mathbf{K}$, allowing for a weak-coupling analysis
of the PDW instability. We find that repulsive electronic interactions combine
to yield effective attractive interactions in the singlet and triplet PDW
channels, as long as the $\Gamma$ pocket is present. Because these PDW channels
decouple from the uniform superconducting channel, they can become the leading
unconventional pairing instability of the system. Upon solving the linearized
gap equations, we find that the PDW instability is robust against small
trigonal warping of the $\pm K$ pockets and small detuning between the $\Gamma$
and $\pm K$ pockets, which affect the PDW transition in a similar way as the
Zeeman magnetic field affects the uniform superconducting transition. We also
derive the Ginzburg-Landau free energy for the PDW gaps with momenta $\pm
\mathbf{K}$, analyzing the conditions for and consequences of the emergence of
FF-type and LO-type PDW ground states. Our classification of the induced orders
in each ground state reveals unusual phases, including an odd-frequency
charge-$2e$ superconductor in the LO-type PDW.
