= Solution
The second-derivative matrix is
$$
\begin{pmatrix}6x&3\lambda z&3\lambda y\\3\lambda z&6y&3\lambda x\\3\lambda y&3\lambda x&6z\end{pmatrix}.
$$
Its determinant is $54[(4+\lambda^3)xyz-\lambda^2(x^3+y^3+z^3)]$. Hence the <Hessian curve> has equation
$$
\boxed{\lambda^2(x^3+y^3+z^3)-(\lambda^3+4)xyz=0.}
$$
If $\lambda=0$, this is the union $xyz=0$ of three lines and is singular. If $\lambda\ne0$, divide by $\lambda^2$ and write it as a Hesse cubic with parameter $\mu=-(\lambda^3+4)/(3\lambda^2)$. The preceding singularity criterion gives $\mu^3=-1$, equivalently
$$
(\lambda^3+4)^3-27\lambda^6=(\lambda^3+1)(\lambda^3-8)^2=0.
$$
Thus the complete condition is
$$
\boxed{\operatorname{Hess}(C)\text{ singular}\iff\lambda=0\text{ or }\lambda^3=-1\text{ or }\lambda^3=8.}
$$
The other values beyond the source's first family are zero and the three cube roots of eight.
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