# Question about Euler's equation for rigid bodies

I wished to understand a particular case of Euler's equation applied in the following cylindrical body: where $$I_{1,2,3}$$ are the moments of inertia. By symmetry, $$I_1=I_2=I_T$$.

Here I consider that no external forces are applied to the cylinder. Stating the equations by their components, we have that:

$$I_1\dot{\omega}_1−(I_2−I_3)\omega_2\omega_3=0$$ $$I_2\dot{\omega}_2−(I_3−I_1)\omega_3\omega_1=0$$ $$I_3\dot{\omega}_3−(I_1−I_2)\omega_1\omega_2=0$$

where $$\omega_i$$ is the rotation speed on axis $$\hat{b}_i$$.

My question is the following: Considering that initially the cylinder spins with speed $$\omega_3$$, and it also moves around axis $$\hat{b}_1$$ and/or $$\hat{b}_2$$, that is, $$\omega_1$$ and/or $$\omega_2$$ are different than zero. According to the third equation, $$\omega_3$$ should remain constant, because $$I_1−I_2=0$$. But I imagine that to conserve its momentum, as the cylinder is rotated to a "laid" position around axis $$\hat{b}_1$$ or $$\hat{b}_2$$, it would keep spinning around $$\hat{b}_3$$, but with a different speed because $$I_3\neq I_T$$.

Am I interpreting this equation wrong?