  We show that the current thermodynamic measurements in the superconducting
phase of $\mathrm{U}\mathrm{Ru}_2\mathrm{Si}_2$ are compatible with two
distinct singlet chiral paired states $k_z(k_x \pm i k_y)$ and $(k_x \pm i
k_y)^2$. Despite possessing similar low temperature thermodynamic properties,
these two pairings are topologically distinguished by their respective orbital
angular momentum projections along the c-axis, $m=\pm 1$ and $m=\pm 2$. The
point nodes of these states act as the monopoles and the anti-monopoles of the
Berry's gauge flux of charge $\pm m$, which are separated in the momentum space
along the $c$ axis. As a result, the Berry's flux through the $ab$ plane equals
$m$. Consequently, the point nodes of $k_z(k_x+i k_y)$ and $(k_x \pm ik_y)^2$
states respectively realize the Weyl and the double-Weyl fermions, with
chemical potential exactly tuned at the Fermi point, due to the charge
conjugation symmetry. These topologically nontrivial point nodes, give rise to
$m$ copies of protected spin degenerate, chirally dispersing surface states on
the $ca$ and the $cb$ planes, which carry surface current, and their energies
vanish at the Fermi arcs. In contrast, a line node acts as the momentum space
vortex loop, and gives rise to the zero energy, dispersionless Andreev bound
states on the surfaces parallel to the plane enclosed by the line node. The
Berry's flux through the $ab$ plane gives rise to anomalous spin Hall and
thermal Hall conductivities, and various magnetoelectric effects. A clear
determination of the bulk invariant can only be achieved by probing the pairing
symmetry via a corner Josephson junction measurement, and Fourier transformed
STM measurements of the Fermi arcs. Therefore, we identify
$\mathrm{U}\mathrm{Ru}_2\mathrm{Si}_2$ as a promising material for realizing
gapless topological superconductivity in three spatial dimensions.
