SOLUTION

$\begin{array}{rcl}x&=&u\, \cos (\omega t) - v \, \sin (\omega t) \\ y &=& u\, \sin (\omega t) + v \, \cos (\omega t) \\ z&=&w \end{array}$ ,

$\begin{array}{rcl}u&=&x\, \cos (\omega t) + y \, \sin (\omega t) \\ v &=& x\, \sin (\omega t) -y \, \cos (\omega t) \\ w&=&z \end{array}$ ,