Euler's equations for constant torque?

[Luftwaff1e]

Homework Statement

An axially symmetric space station (principal axis e₃, and λ₁ = λ₂) is floating in free space. It has rockets mounted symmetrically on either side that are firing and exert a constant torque τ about the symmetry axis. Solve Euler's equations exactly for ω (relative to the body axis) and describe the motion. At t = 0 take ω = (ω₁,0,ω₃).

As well I am told to introduce the constants
(λ₃-λ₁)ω₃/λ₁

α = (λ₃-λ₁)τ/(λ₁λ₃)

Homework Equations

n=ω₁+iω₂

The Attempt at a Solution

I tried several different things trying to solve the problem but unfortunately I could never quite figure out which parts of Euler's equation are zero. I think that the third τ in Eulers equation is zero which would make ω₃ constant but then I'm not really sure where to go.

 

[TSny]

x₃ is the symmetry axis. The problems states that the torque τ is about this axis. That means that \Gamma₃ = τ