  We report the discovery of structural phase transitions and superconductivity
in the full Heusler compounds $X$Pd$_2$Sn ($X$ = Ti, Zr, Hf), by means of
electrical transport, magnetic susceptibility, specific heat and x-ray
diffraction measurements. TiPd$_2$Sn, ZrPd$_2$Sn and HfPd$_2$Sn undergo
structural phase transitions from the room-temperature cubic MnCu$_2$Al-type
structure (space group $Fm\bar{3}m$) to a low-temperature tetragonal structure
at around 160 K, 110 K and 90 K, respectively, which are likely related charge
density wave (CDW) instabilities. Low temperature single crystal x-ray
diffraction measurements of ZrPd$_2$Sn demonstrate the emergence of a
superstructure with multiple commensurate modulations below $T_s$. ZrPd$_2$Sn
and HfPd$_2$Sn have bulk superconductivity (SC) with transition temperatures
$T_c$ $\sim$ 1.2 K and 1.3 K, respectively. Density functional theory (DFT)
calculations reveal evidence for structural and electronic instabilities which
can give rise to CDW formation, suggesting that these $X$Pd$_2$Sn systems are
good candidates for examining the interplay between CDW and SC.
