Yes it is possible to have a hyperspectral and lidar system to work simultaneously on the same airborne platform. In fact NOAA has conducted a series of flights as part of the National Geodetic Survey project where a fusion of hyperspectral, lidar and high resolution imaging were employed simultaneously in the same aircraft. The name of the aircraft is the NOAA Cessna Citation which is used in a number of Remote Sensing operations. It was also employed on various other occasions following 9/11, post hurricane damage mapping after Katrina, Wilma etc. and continues to do so.
Other examples are the Carnegie Airborne Observatory (CAO) which has collected data in South Africa, https://cao.carnegiescience.edu/ and NASA Goddard's Lidar, Hyperspectral and Thermal (G-LiHT), gliht.gsfc.nasa.gov/
thank you all for your contributions, a follow up question will be how can radiometric corrections be performed on such a system considering the different settings on the hyperspectral and the Lidar scanners?