Solution
= Solution
A <streamline> $x(s)=(x(s),y(s),z(s))$ satisfies
$$
\dot x=\alpha y,
\qquad
\dot y=-\alpha x,
\qquad
\dot z=\beta,
$$
with $(x(0),y(0),z(0))=(1,0,0)$. Therefore
$$
\boxed{x(s)=\cos(\alpha s),\qquad
y(s)=-\sin(\alpha s),\qquad
z(s)=\beta s}.
$$
For $\beta\ne0$ this is a helix, with $x=\cos(\alpha z/\beta)$ and $y=-\sin(\alpha z/\beta)$; for $\beta=0$ it is the unit circle in the plane $z=0$.