= Solution
Put $r=|\boldsymbol\theta|$ and write the vector <thin gravitational lens equation> as $\boldsymbol\beta=(1-\bar\kappa(r))\boldsymbol\theta$. A tangential displacement at fixed radius has <eigenvalue> $\lambda_t=1-\bar\kappa$. A radial displacement has <eigenvalue> $\lambda_r=1-\alpha'(r)$. From the definition of the mean <lensing convergence>,
$$
r^2\bar\kappa(r)=2\int_0^r\kappa(s)s\,ds,
\qquad 2\bar\kappa+r\bar\kappa'=2\kappa.
$$
Since $\alpha=r\bar\kappa$, this gives $\alpha'=2\kappa-\bar\kappa$. Consequently,
$$
\det A=(1-\bar\kappa)(1-2\kappa+\bar\kappa).
$$
The two requested <gravitational-lensing critical curve> conditions are therefore
$$
\boxed{\bar\kappa=1\quad\text{or}\quad 2\kappa-\bar\kappa=1.}
$$
They correspond respectively to a <tangential critical curve of an axisymmetric lens> and a <radial critical curve of an axisymmetric lens>.
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