= Solution
The phase-plane equations are $a'=b$, $b'=-cb-f(a)$. Nonnegative equilibria require $b=0$ and $f(a)=0$, hence only $\boxed{(0,0),\ (\eta,0)}$. At the origin $f'(0)=\eta^2$, so the linear <eigenvalues> solve
$$
\lambda^2+c\lambda+\eta^2=0,\qquad
\boxed{\lambda_\pm=(-c\pm\sqrt{c^2-4\eta^2})/2.}
$$
For $0<c<2\eta$ the roots are complex with negative real part, giving a <stable focus>. For $c>2\eta$ both are negative real numbers, giving a <stable node>. Equality is the repeated-root threshold and is not covered by either strict classification.
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