Conical refraction (CR) is observed for the light propagating along the optical axis of a biaxial crystal. In this case a narrow beam evolves as a hollow double-walled cylinder of light behind the exit facet of a crystal. Despite of almost two-centuries-long research, CR was slow with practical applications, mainly due to the difficulties associated with cutting of the biaxial crystals with the necessary precision. However, a number of recent papers report on the emerging applications of CR for the realisation of ultra-efficient CR lasers, lasers with CR output, optical trapping with CR beams, utilisation of CR for quantum-computing, cryptography and super-resolution microscopy (see e.g.  and references therein). However, one of the most novel and intriguing phenomena within the CR are associated with utilization of vortex input beams. These are capable to completely change the familiar CR patterns  and trigger many new applications. In this sense, studies of CR with Laguerre-Gaussian beams LGl0, which are the simplest vortices, may be very fruitful (here, I is the index determining vortex charge): These already have demonstrated specific new properties of CR, such as formation of a multi-ring image in the Lloyd plane .