= Solution
The <Jacobian matrix> is
$$
J(x,y)=
\begin{pmatrix}
y-k-6x+3x^2 & x\\
-y & 2y-1-x
\end{pmatrix}.
$$
At $(0,0)$ its <eigenvalues> are $-k$ and $-1$. It is therefore a stable node for $k>0$, a saddle for $k<0$, and nonhyperbolic at $k=0$, consistent with the proposed transcritical bifurcation.
At $(0,1)$ the matrix is
$$
\begin{pmatrix}1-k&0\\-1&1\end{pmatrix},
$$
with eigenvalues $1-k$ and $1$. It is an unstable node for $k<1$, a saddle for $k>1$, and nonhyperbolic at $k=1$, again consistent with a transcritical bifurcation.
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