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 $d(k)=(b + ic)(k_b + ik_c)$.
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