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Previous works have been devoted to the symmetry properties of the Poynting vector for longitudinal axisymmetric beams expressed in terms of beam shape coefficients (BSCs). In the present paper, we examine the case of circularly axisymmetric beams. The former is defined as beams whose longitudinal component of the Poynting vector does not depend on the azimuthal angle. The latter is defined as beams whose all components of the Poynting vector do not depend on the azimuthal angle. We establish that two previously studied families of longitudinal axisymmetric beams are actually circularly axisymmetric beams as well. Examples are provided. • Symmetries of Poynting vector may be expressed in terms of beam shape coefficients of the GLMT. • Such symmetries are studied in the case of circularly symmetric beams. • Two families of such circularly symmetric beams are examined. • Examples and counter-examples are provided. [ABSTRACT FROM AUTHOR]