  We study the quasiparticle excitations around a single vortex in the
superconducting topological insulator Cu$_{x}$Bi$_{2}$Se$_{3}$, focusing on a
superconducting state with point nodes. Inspired by the recent Knight shift
measurements, we propose two ways to detect the positions of point nodes, using
an explicit formula of the density of states with Kramer-Pesch approximation in
the quasiclassical treatment. The zero-energy local density of states around a
vortex parallel to the $c$-axis has a twofold shape and splits along the nodal
direction with increasing energy; these behaviors can be detected by the
scanning tunneling microscopy. An angular dependence of the density of states
with a rotating magnetic field on the $a$-$b$ plane has deep minima when the
magnetic field is parallel to the directions of point nodes, which can be
detected by angular-resolved heat capacity and thermal conductivity
measurements. All the theoretical predictions are detectable via standard
experimental techniques in magnetic fields.
