  We describe the Harder--Narasimhan filtration of the Hodge bundle for
Teichm\"uller curves in the non-varying strata of quadratic differentials
appearing in [CM2]. Moreover, we show that the Hodge bundle on the non-varying
strata away from the irregular components can split as a direct sum of line
bundles. As applications, we determine all individual Lyapunov exponents of
algebraically primitive Teichm\"uller curves in the non-varying strata and
derive new results regarding the asymptotic behavior of Lyapunov exponents.
