In Nature there is no so small thing which, by intend look turned to it, would not grow up to infinity L. N. Tolstoi
I. INTRODUCTIONRecently the general linear transformation for point rotation coordinate frames was considered [1]. A distinguishing feature of the frame, in contrast to the Cartesian one, is the existence of the rotation axis at every point. The frame coordinates are an angle and time, the frequency of rotation is a parameter. The concept of the frame originated from the optical indicatrix (index ellipsoid) [2]. Rotation of the optical indicatrix arises in three-fold electrooptical crystals under the action of the rotating electric field applied perpendicular to the optical axis [3]. Such a rotation is possible as in the Pockels as Kerr crystals and also in the isotropic Kerr medium. The rotation is used in single-sideband modulators [4], [5]. The single-sideband modulation has very interesting features from the theoretical viewpoint. In applications it may be used for the frequency modulation and frequency shifting. In contrast to usual modulation such a shifting is "100% transformation" of the initial into output frequency. However at present the modulation practically is not in use. It is connected with the high controlling voltage of bulk modulators [5]; creating waveguide single-sideband modulators calls for considerable technological efforts [6].Two point are essential by the consideration of a plane circularly polarized wave propagating through a medium with the rotating optical indicatrix. That is the necessity to use the non-Cartesian point rotation frame and the necessity to know what is the frequency superposition law by the transition from one rotating frame to another. The usual description in the Cartesian frame tacitly assumes that this law is Galilean or linear one. In this law the frequency of any field may be infinitely large.In the general case considered in [1] the reverse frequency, i.e., the frequency of the second frame relative to the first one, is a function of the direct frequency. However both the frequencies are assumed to be symmetric, i.e., the direct frequency is the same function of the reverse frequency. Using symmetry of the transformation under interchanging coordinates and assuming that this function is kept by such a interchange, it was shown that three different types of the transformation are possible. The first type is a generalization of the Lorentz transformation. The second and third types are principally different and possess unusual properties, in particular, an uncertainty of time determination and solutions with lower and upper frequency boundaries.The point rotation frames have not transverse coordinates, however a coordinate along the axis of rotation can be used as the space coordinate. The approach developed in [1] for two-dimensional (2D) case is unacceptable for the 3D case. In this paper we somewhat modify the approach using again only symmetry. The main idea of the modification is a "velocity invariant" which remains unchanged by replacing any ...