  Topological superconductivity is a long-sought state of matter in bulk
materials, and odd-parity superconductor UTe$_2$ is a prime candidate. The
recent observation of a field-trainable spontaneous Kerr signal in UTe$_2$ at
the onset of superconductivity provides strong evidence that the
superconducting order parameter is multicomponent and breaks time-reversal
symmetry. Here, we perform Kerr effect measurements on a number of UTe$_2$
samples -- grown $via$ both chemical vapor transport and the molten-salt-flux
methods -- that show a single superconducting transition between 1.6~K and
2.1~K. Our results show no evidence for a spontaneous Kerr signal in zero field
measurements. This implies that the superconducting state of UTe$_2$ does not
intrinsically break time-reversal symmetry. Instead, we observe a
field-trainable signal that varies in magnitude between samples and between
different locations on a single sample, which is a sign of inhomogeneous
magnetic regions. Our results provide an examination of representative UTe$_2$
samples and place strong constraints on the superconducting order parameter of
UTe$_2$.
