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We studied how the application of an in-plane field affects the asymmetries of the domain nucleation activity and domain wall velocity revealed previously in ultrathin Co films with perpendicular anisotropy. It is established that the asymmetries survive and new effects arise under application of an additional field parallel to the film surface. It is found that the mobility of different parts of a circular domain wall vary drastically under application of additional planar field. The domain wall velocity in ultrathin Co films, unlike that in low-damping garnet films, slows down dramatically upon application of a planar field. The domain wall part that is moving perpendicular to the in-plane field direction decelerates most relative to the other parts of the domain wall. Moreover, domain wall parts moving in the same direction as the in-plane field and in the opposite direction have considerably different velocities. This anisotropy of domain wall velocity rotates with the field as the in-plane field direction is varied. A new asymmetry in the domain wall velocity in the presence of both in-plane and perpendicular fields is observed when the perpendicular field is reversed.