  In many cases, unconventional superconductivity are realized by suppressing
another order parameter, such as charge density wave (CDW) or spin density wave
(SDW). This suggests that the fluctuations of these order parameters play an
important role in producing superconductivity. LaPt$_2$Ge$_2$ undergoes a
tetragonal-to-monoclinic structural phase transition (SPT) at $T_{\rm s}$ = 394
K, accompanying a double period modulation in the $a$-axis direction, and
superconducts at $T_{\rm c}$ = 0.41 K. We performed band calculations and found
2D (two dimensional)-like Fermi surfaces with partial nesting. A reduction in
the density of states in the monoclinic phase was found in the calculation and
confirmed by $^{195}$Pt-NMR. We suggest a CDW as a possible cause for the SPT.
By changing the stoichiometry between Pt and Ge, we succeeded in suppressing
$T_{\rm s}$ and increasing $T_{\rm c}$ in LaPt$_{2-x}$Ge$_{2+x}$. Comparison of
$^{139}$La- and $^{195}$Pt-NMR data reveals moderate fluctuations associated
with SPT. From $^{139}$La-NQR measurements at zero field, we found that an
isotropic superconducting gap is realized in LaPt$_{2-x}$Ge$_{2+x}$ ($x$ =
0.20). We discuss the relationship between superconductivity and the SPT
order/fluctuations.
