  We develop the embedded gradient vector field method, introduced in [8] and
[9], for the case of the special unitary group $\mathcal{SU}(N)$ regarded as a
constraint submanifold of the unitary group $\mathcal{U}(N)$. The optimization
problem associated to the trace fidelity cost function defined on
$\mathcal{SU}(N)$ that appears in the context of $\mathcal{SU}(N)$ quantum
control landscapes is completely solved using the embedded gradient vector
field method. We prove that for $N\geq 5$, the landscape is not
$\mathcal{SU}(N)$-trap free, there are always kinematic local extrema that are
not global extrema.
