Since there is one equation and two unknowns, it must be possible to define, say, beta in terms of alpha, where alpha can be anything. For G = 0 we have
(-Om^3*t + (a+b)*Om)^2 = -(Om^2 - a*f)^2
-Om^3*t + (a+b)*Om = +-*i*(Om^2 - a*f)
where there are obvious notational substitutions for Omega, tau, alpha, beta, and the +- choice gives two different solutions. Solving for b,
b = (1/Om)*( Om^3*t -a*Om +-i*(Om^2 - a*f) )
where 'a' can be anything. Solving instead for a (this does not give a different family of solutions, rather the same ones expressed differently) gives
a = (Om^3*t +-i*Om^2 -b*Om)/(Om +-i*f)
Here the sign in the denominator (+ or -) has to match the sign in the denominator, and b can be anything. The choice b = 0 gives the solutions from Star Strider.