= Solution
Choose any nonzero auxiliary <vector> $\mathbf q$ not collinear with $\mathbf p$. Then $\mathbf p,\mathbf q,\mathbf p\times\mathbf q$ form a <basis>, so write
$$
\mathbf x=\alpha\mathbf p+\beta\mathbf q+\gamma(\mathbf p\times\mathbf q).
$$
Taking the <dot product> with $\mathbf p$ gives $\alpha|\mathbf p|^2+\beta\mathbf p\cdot\mathbf q=k$. This separates the two cases requested in the hint.
If $\mathbf p\cdot\mathbf q\ne0$, choose $\alpha,\gamma$ freely and solve for $\beta$:
$$
\boxed{\mathbf x=\alpha\mathbf p+
\frac{k-\alpha|\mathbf p|^2}{\mathbf p\cdot\mathbf q}\,\mathbf q
+\gamma(\mathbf p\times\mathbf q),\qquad \alpha,\gamma\in\mathbb R.}
$$
If $\mathbf p\cdot\mathbf q=0$, the coefficient of $\mathbf p$ is fixed and the other two are free:
$$
\boxed{\mathbf x=\frac{k}{|\mathbf p|^2}\mathbf p
+\beta\mathbf q+\gamma(\mathbf p\times\mathbf q),\qquad\beta,\gamma\in\mathbb R.}
$$
These formulas describe the whole <plane> with normal $\mathbf p$, including $k=0$. Completeness follows from the <basis> expansion, not only from checking that the displayed <vectors> satisfy the equation. The <parametrization of a plane with an auxiliary vector> also gives a single formula valid in both cases.
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