The Golden Ratio shows up in the cosine function. Do other related numbers do so as well?
Substitute $x=m \cos t$ to get $$m^3 \cos^3 t - a m \cos t -1=0$$ Let $$m=\sqrt{\frac{4a}{3}}$$ then it rearranges (assuming $a,m\ne 0$) to $$4 \cos^3 t - 3 \cos t = \sqrt{\frac{27}{4a^3}}$$ Using trigonometric identity $$\cos (3t) = \sqrt{\frac{27}{4a^3}}$$ So $$x = \sqrt{\frac{4a}{3}} \cos\left(\frac{1}{3} \cos^{-1}\left(\sqrt{\frac{27}{4a^3}}\right)\right)$$
When $a=1$ the R.H.S. of the penultimate equation is bigger than $1$ so $\cos^{-1}$ is problematic. Note that $\cos^{-1}/3$ is multi-valued.