1. For the system of inequalities about real numbers $x, y$
$$
\left\{\begin{array}{l}
y \geqslant x^{2}+2 a, \\
x \geqslant y^{2}+2 a
\end{array}\right.
$$

to have a unique solution, the values of the parameter $a$ are
1. $\frac{1}{8}$.

From the problem, we know that the parabolas
$$
y=x^{2}+2 a \text { and } x=y^{2}+2 a
$$

are tangent to each other, and $y=x$ is their common tangent line.
Therefore, $a=\frac{1}{8}$.