  We present comprehensive investigations into the structural, superconducting,
and topological properties of Bi$_2$PdPt. Magnetization and heat capacity
measurements performed on polycrystalline Bi$_2$PdPt demonstrate a
superconducting transition at $\approx$ 0.8 K. Moreover, muon spin
relaxation/rotation ($\mu$SR) measurements present evidence for a time reversal
symmetry preserving, isotropically gapped superconducting state in Bi$_2$PdPt.
We have also performed density-functional theory (DFT) calculations on
Bi$_2$PdPt alongside the more general isostructural systems,
BiPd$_{x}$Pt$_{1-x}$, of which Bi$_2$PdPt and $\gamma$-BiPd are special cases
for $x=0.5$ and $x=1$ respectively. We have calculated the $Z_2$ topological
index from our DFT calculations for a range of substitution fractions, $x$,
between $x=0$ and $x=1$ characterizing the topology of the band structure. We
find a non-trivial topological state when $x>0.75$ and a trivial topological
state when $x<0.75$. Therefore our results indicate that BiPd$_{x}$Pt$_{1-x}$
could be a topological superconductor for $x>0.75$.
