Laser beams whose complex amplitudes can be described by explicit analytical relationships that are exact solutions of the paraxial propagation equation have invariably been the focus of attention of optical researchers. The fact is that knowledge of the complex amplitude of the beams at any distance from the source plane enables the full set of their parameters to be predicted, including the intensity distribution, intensity moments, total power of the beam, topological charge, orbital angular momentum, and beam divergence. When propagated in free space, such beams are normally either structurally stable (retaining their intensity pattern up to a scale and rotation) or weakly changing, with their complex amplitude expressed by an elegant mathematical relationship. While such beams are abound, the most familiar are Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) laser beams. For the first time, conventional HG beams were discussed in Ref. (Kogelnik and Li in Proc. IEEE 54:1312–1329, 1966) and elegant HG beams were studied in Ref. (Siegman in J. Opt. Soc. Am. 63:1093–1094, 1973). Various forms of generalized HG beams have also been discussed in Refs. (Pratesi and Ronchi in J. Opt. Soc. Am. 67:1274–1276, 1977; Zauderer in J. Opt. Soc. Am. A 3:465–469, 1986; Wünsche in J. Opt. Soc. Am. A 6:1320–1329, 1989; Kotlyar and Kovalev in J. Opt. Soc. Am. A 31:274–282, 2014; Wang et al. in J. Opt. 18, 2016). More specifically, in Ref. (Pratesi and Ronchi in J. Opt. Soc. Am. 67:1274–1276, 1977) the generalized HG beams were defined as conventional HG beams with a parameter, whereas in Ref. (Wang et al. in J. Opt. 18, 2016) the generalized beams were obtained as superposition of conventional HG beams propagating with the same phase velocity. Combined Hermite–Laguerre-Gaussian beams proposed in Ref. (Abramochkin and Volostnikov in J. Opt. A Pure Appl. Opt. 6:S157–S161, 2004) were shown to transform into conventional HG beams or LG beams depending on the specific value of the parameter. A relationship between the HG and LG beams was established in Ref. (Abramochkin and Volostnikov in Opt. Commun. 83:123–135, 1991) and vortex HG beams constructed as superposition of conventional HG beams with phase shifts were studied in Ref. (Kotlyar et al. in Opt. Lett. 40:701–704, 2015). For the first time, the LG beams were analyzed as laser cavity modes in Ref. (Kogelnik and Li in Appl. Opt. 5:1550–1567, 1996) before being shown to propagate in an ABCD system (Mei et al. in Optik 115:311–316, 2004). Issues relating to the study of LG modes included their transformation and focusing with the aid of an axicon (Arlt et al. in J. Mod. Opt. 48:783–787, 2001; Jarutis et al. in Opt. Commun. 184:105–112, 2000) and their representation through Wigner functions (Simon and Agarwal in Opt. Lett. 25:1313–1315, 2000). Non-paraxial variants of the HG and LG beams have also been analyzed (Kim and Lee in Opt. Commun. 169:9–16, 1999), as well as studying double-frequency LG beams and their transformation into conventional LG beams (Hasegawa and Shimizu in Opt. Commun. 160:103–108, 1999). Other types of beams studied, included asymmetric LG beams (Kovalev et al. in Phys. Rev. A 93, 2016) and laser-aided intra-cavity shaping of asymmetric HG beams (Hsieh et al. in Opt. Express 26:31,738–31,749, 2018), which were then transformed into asymmetric LG beams using a mode converter (Abramochkin and Volostnikov in Opt. Commun. 83:123–135, 1991). A technique for generating vector laser beams with the aid of LG beams has been proposed (Goran Abad and Mahmoudi in Sci. Rep. 11:5972, 2021). Generation of LG beams in a wide frequency band by second harmonic generation has been reported (Yang et al. in J. Opt. 51:910–926, 2022), whereas conventional LG beams can be realized using a standard spatial light modulator (SLM) (Matsumoto et al. in J. Opt. Soc. Am. A 25:1642–1651, 2008).