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\item[Background] For cranked mean-field calculations with arbitrarilyoriented rotational frequency vector $\boldsymbol{\omega}$ in the intrinsicframe, one has to employ constraints on average values of the quadrupole-momenttensor, so as to keep the nucleus in the principal-axis reference frame. Kermanand Onishi [Nucl. Phys. A {\bf 361}, 179 (1981)] have shown that the Lagrangianmultipliers that correspond to the required constraints are proportional to$\boldsymbol{\omega} \times \boldsymbol{J}$, where $\boldsymbol{J}$ is theaverage angular momentum vector.  \item[Purpose] We study the validity and consequences of the Kerman-Onishiconditions in the context of self-consistent tilted-axis-cranking (TAC)mean-field calculations.  \item[Methods] We perform a two-dimensional self-consistent calculations(with and without pairing) utilizing the symmetry-unrestricted solver {\schfodd}. At each tilting angle, we compare the calculated values ofquadrupole-moment-tensor Lagrangian multipliers and $\boldsymbol{\omega} \times\boldsymbol{J}$.  \item[Results] We show that in self-consistent calculations, theKerman-Onishi conditions are obeyed with high precision. Small deviations seenin the calculations with pairing can be attributed to the truncation of thequasiparticle spectrum. We also provide results of systematic TAC calculationsfor triaxial strongly deformed bands in $^{160}$Yb.  \item[Conclusions] For non-stationary TAC solutions, Kerman-Onishi conditionslink the non-zero values of the angle between rotational-frequency andangular-momentum vectors to the constraints on off-diagonal components of thequadrupole-moment tensor. To stabilize the convergence of self-consistentiterations, such constraints have to be taken into account. Only then one candetermine the Routhian surfaces as functions of the tilting angles.

Kerman-Onishi conditions in self-consistent tilted-axis-cranking mean-field calculations