Attachment sheath for an accessory, the sheath comprising: - an axial rectilinear structure comprising at least an outer surface and an inner surface, the inner surface defining a housing, this housing comprising an orifice at at least a first end of the rectilinear structure, - at least one cooperation interface comprising at least a first portion fastened to the outer surface of the rectilinear structure and a second portion, the end of which forms at least three non-aligned bearing points, the cooperation interface being configured to position the axis of the rectilinear structure in a plane parallel to the plane defined by the at least three non-aligned bearing points of the cooperation interface, the centre of gravity of the attachment sheath being positioned such that the axis passing through this centre of gravity and perpendicular to the plane defined by the at least three non-aligned bearing points passes through this plane in a region delimited by these at least three non-aligned bearing points, said sheath being characterized in that, in a cross-sectional plane comprising the axis of the rectilinear structure and perpendicular to the plane defined by the at least three non-aligned bearing points of the cooperation interface, at least the second end of the rectilinear structure comprises an arrangement forming a continuous inclination between, on the one hand, the outer surface portion of the rectilinear structure positioned facing the end of the bearing points and, on the other hand, the outer surface portion of the rectilinear structure that is opposite thereto with respect to the axis of the rectilinear structure such that the distance between, on the one hand, the second end of the rectilinear structure and, on the other hand, the perpendicular to the axis of the rectilinear structure passing through the fastening of the cooperation interface on the outer surface of the rectilinear structure has a greater length at the outer surface portion of the rectilinear structure positioned facing the end of the bearing points than at the outer surface portion of the rectilinear structure that is opposite thereto. - an axial rectilinear structure comprising at least an outer surface and an inner surface, the inner surface defining a housing, this housing comprising an orifice at at least a first end of the rectilinear structure, - at least one cooperation interface comprising at least a first portion fastened to the outer surface of the rectilinear structure and a second portion, the end of which forms at least three non-aligned bearing points, the cooperation interface being configured to position the axis of the rectilinear structure in a plane parallel to the plane defined by the at least three non-aligned bearing points of the cooperation interface, the centre of gravity of the attachment sheath being positioned such that the axis passing through this centre of gravity and perpendicular to the plane defined by the at least three non-aligned bearing points passes through this plane in a region delimited by these at least three non-aligned bearing points, said sheath being characterized in that, in a cross-sectional plane comprising the axis of the rectilinear structure and perpendicular to the plane defined by the at least three non-aligned bearing points of the cooperation interface, at least the second end of the rectilinear structure comprises an arrangement forming a continuous inclination between, on the one hand, the outer surface portion of the rectilinear structure positioned facing the end of the bearing points and, on the other hand, the outer surface portion of the rectilinear structure that is opposite thereto with respect to the axis of the rectilinear structure such that the distance between, on the one hand, the second end of the rectilinear structure and, on the other hand, the perpendicular to the axis of the rectilinear structure passing through the fastening of the cooperation interface on the outer surface of the rectilinear structure has a greater length at the outer surface portion of the rectilinear structure positioned facing the end of the bearing points than at the outer surface portion of the rectilinear structure that is opposite thereto.