= Solution
Write
$$
\mathbf x=\mathbf a+\mathbf u,
\qquad
\mathbf y=\mathbf a-\mathbf u.
$$
The first equation is then automatic, while the second becomes
$$
\mathbf x\cdot\mathbf y
=\mathbf a\cdot\mathbf a-\mathbf u\cdot\mathbf u=c.
$$
Thus the general solution is
$$
\boxed{\mathbf x=\mathbf a+\mathbf u,
\qquad \mathbf y=\mathbf a-\mathbf u,
\qquad |\mathbf u|=\sqrt{\mathbf a\cdot\mathbf a-c}}.
$$
As $\mathbf u$ varies, $\mathbf x$ and $\mathbf y$ are antipodal points of the <sphere> with center $\mathbf a$ and radius $\sqrt{\mathbf a\cdot\mathbf a-c}$, so they are opposite ends of a diameter.
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