  The gap structure of a novel uranium-based superconductor UTe$_2$, situated
in the vicinity of ferromagnetic quantum criticality, has been investigated via
specific-heat $C(T,H,\Omega)$ measurements in various field orientations. Its
angular $\Omega(\phi,\theta)$ variation shows a characteristic shoulder anomaly
with a local minimum in $H \parallel a$ at moderate fields rotated within the
$ab$ and $ac$ planes. Based on the theoretical calculations, these features can
be attributed to the presence of point nodes in the superconducting gap along
the $a$ direction. Under the field orientation along the easy-magnetization $a$
axis, an unusual temperature dependence of the upper critical field at low
fields together with a convex downward curvature in $C(H)$ were observed. These
anomalous behaviors can be explained on the basis of a nonunitary triplet state
model with equal-spin pairing whose $T_{\rm c}$ is tuned by the magnetization
along the $a$ axis. From these results, the gap symmetry of UTe$_2$ is most
likely described by a vector order parameter of $d(k)=(b + ic)(k_b + ik_c)$.
