Vortex states in an unconventional superconductor and the mixed phases of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">UPt</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>

Recent experiments on ${\mathrm{UPt}}_{3}$ indicate the existence of more than one superconducting phase and an associated unconventional order parameter belonging to a nontrivial representation of the crystal point group ${\mathit{D}}_{6\mathit{h}}$. In this article we consider the Ginzburg-Landau (GL) theory for the vortex states in the two-dimensional ${\mathit{E}}_{1}$ representation for the order parameter, the representation believed to describe the phases of ${\mathrm{UPt}}_{3}$. One of the zero-field superconducting phases is predicted to break time-reversal symmetry. We show that the lower critical field for vortex nucleation in a ground state with broken time-reversal symmetry exhibits a novel asymmetry. Observation of this asymmetry in any superconductor would provide clear evidence for an unconventional order parameter. We present solutions to the GL equations for singly quantized vortices. For rectilinear vortices along the c axis two classes of solutions are found: (i) axially symmetric vortices and (ii) nonaxisymmetric vortices which spontaneously break the axial symmetry of the condensate density. We discuss the structure of these vortices, their relation to the asymmetry of ${\mathit{H}}_{\mathit{c}1}$, the possibility of phase transitions between vortex states of different symmetry, and the phase diagram of ${\mathrm{UPt}}_{3}$ in the H-T plane.