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A systematic search for possible order parameters of the so-called hiddenorder of URu$_{2}$Si$_{2}$ is conducted. Among the possible candidates that dofulfill the experimental symmetry restrictions on the hidden order parameter,we find one candidate that stand out -- one of the components of atriakontadipole multipole tensor that belongs to the $A_{1u}$ irreduciblerepresentation of the point group $D_{4h}$. This solution is characterized by a$\vec{Q}=(0,0,1)$ ordering of the triakontadipoles and has a symmetry thatforbids magnetic moments as well as most other multipole ordering on theuranium sites. This hidden order phase is closely related to the theantiferromagnetic phase, which is manifested in the similarities of thegeometries of the calculated Fermi surfaces. Finally is is found that thisnon-magnetic solution allows for another secondary superimposed order parameterthat belong to $B_{2u}$, which gives rise to an anisotropic in-planesusceptibility without breaking the tetragonal crystal symmetry.

Systematic study of the hidden order in URu$_{2}$Si$_{2}$ as a multipolar order; the role of triakontadipoles