Finsleroids with three axes in dimension $N=3$

The Conclusive Theorem has been established to determine the dependence of the three-axes positive-definite Finsleroid metric functions $F$ on the Finsleroid azimuthal angle $\theta$ in the three-dimensional case $N=3$, provided that the condition of the angle-separation in the involved characteristic functions is implied. The complete set of algebraic and differential equations is derived in all rigor which are necessary and sufficient in order that the function $F$ belong to the class. It proves possible to solve the equations and obtain the explicit dependence of the involved characteristic functions on the angle $\theta$.