Although Zhuyan Yu was nominally the acting chief of the frontline headquarters, in reality, there wasn't much for her to worry about. Most predicaments had been more or less resolved before Wang Xu and others returned to Earth. Thus, her existence was merely to maintain order.
Obviously, Zhuyan Yu was aware of this and was not too involved in management, instead, focusing her efforts on researching magic.
With Feng's return, she and Nightingale were eager to unravel the mystery of Feng's abilities, such as how she cast magic formulas, a scenario they had yet to comprehend.
The path for traditional mages to cast magic formulas is clear. They absorb magic power through their bodies, convert it and then release it through a magic wand, their medium. So, the presence of both the mage and the Magic Wand must be constant.