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Some circumstellar disks, called transitional or hybrid disks, present characteristics of both protoplanetary disks (significant amount of gas) and debris disks (evolved structures around young main-sequence stars, composed of second generation dust, from collisions between planetesimals). Therefore, they are ideal astrophysical laboratories to witness the last stages of planet formation. The circumstellar disk around HD 141569A was intensively observed and resolved in the past from space, but also from the ground. However, the recent implementation of high contrast imaging systems has opened up new opportunities to re-analyze this object. We analyzed Gemini archival data from the Near-infrared Coronagraphic Imager obtained in 2011 in the H band, using several angular differential imaging techniques (classical ADI, LOCI, KLIP). These images reveal the complex structures of this disk with an unprecedented resolution. We also include archival Hubble Space Telescope images as an independent data set to confirm these findings. Using an analysis of the inner edge of the disk, we show that the inner disk is almost axisymmetrical. The measurement of an offset toward the east observed by previous authors is likely due to the fact that the eastern part of this disk is wider and more complex in substructure. Our precise reanalysis of the eastern side shows several structures, including a splitting of the disk and a small finger detached from the inner edge to the southeast. Finally, we find that the arc at 250 AU is unlikely to be a spiral, at least not at the inclination derived from the first ring, but instead could be interpreted as a third belt at a different inclination. If the very symmetrical inner disk edge is carved by a companion, the data presented here put additional constraints on its position. The observed very complex structures will be confirmed by the new generation of coronagraphic instrument (GPI, SPHERE). However, a full understanding of this system will require gas observations at millimetric wavelengths. [ABSTRACT FROM AUTHOR]