= Solution
The <linear system of differential equations> has <matrix>
$$
M=\begin{pmatrix}0&\alpha\\-\alpha&0\end{pmatrix},\qquad
C=\begin{pmatrix}1&1\\i&-i\end{pmatrix},\qquad C^{-1}MC=\begin{pmatrix}i\alpha&0\\0&-i\alpha\end{pmatrix}.
$$
The columns of $C$ are <eigenvectors> of $M$. The <change of basis> $v'=C^{-1}v$ therefore gives $v'_1=(v_1-iv_2)/2$, $v'_2=(v_1+iv_2)/2$, with independent <first-order linear differential equations>
$$
\dot v'_1=i\alpha v'_1,\quad \dot v'_2=-i\alpha v'_2,\qquad
v'_1(t)=K_+e^{i\alpha t},\quad v'_2(t)=K_-e^{-i\alpha t}.
$$
The initial <velocity> gives $K_+=-iv_0/2$, $K_-=iv_0/2$. Transforming back, $v=Cv'$, yields the real <velocity> <vector components>
$$
\boxed{v_1(t)=v_0\sin\alpha t,\qquad v_2(t)=v_0\cos\alpha t.}
$$
These also cover $\alpha=0$. Complex <coordinates> have decoupled a real rotation without making the physical <velocity> complex.
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