  Quantum systems with dynamical symmetries have conserved quantities which are
preserved under coherent controls. Therefore such systems can not be completely
controlled by means of only coherent control. In particular, for such systems
maximal transition probability between some pair of states over all coherent
controls can be less than one. However, incoherent control can break this
dynamical symmetry and increase the maximal attainable transition probability.
Simplest example of such situation occurs in a three-level quantum system with
dynamical symmetry, for which maximal probability of transition between the
ground and the intermediate state by only coherent control is $1/2$, and by
coherent control assisted by incoherent control implemented by non-selective
measurement of the ground state is about $0.687$, as was previously
analytically computed. In this work we study and completely characterize all
critical points of the kinematic quantum control landscape for this
measurement-assisted transition probability, which is considered as a function
of the kinematic control parameters (Euler angles). This used in this work
measurement-driven control is different both from quantum feedback and
Zeno-type control. We show that all critical points are global maxima, global
minima, saddle points and second order traps. For comparison, we study the
transition probability between the ground and highest excited state, as well as
the case when both these transition probabilities are assisted by incoherent
control implemented by measurement of the intermediate state.
