  Quantum discord Q is a function of density matrix elements. The domain of
such a function in the case of two-qubit system with X density matrix may
consist of three subdomains at most: two ones where the quantum discord is
expressed in closed analytical forms (Q_{\pi/2} and Q_0) and an intermediate
subdomain for which, to extract the quantum discord Q_\theta, it is required to
solve in general numerically a one-dimensional minimization problem to find the
optimal measurement angle \theta\in(0,\pi/2). Hence the quantum discord is
given by a piecewise-analytic-numerical formula Q=\min{Q_{\pi/2}, Q_\theta,
Q_0}. Equations for determining the boundaries between these subdomains are
obtained. The boundaries consist of bifurcation points. The Q_{\theta}
subdomains are discovered in the generalized Horodecki states, in the dynamical
phase flip channel model, in the anisotropic spin systems at thermal
equilibrium, in the heteronuclear dimers in an external magnetic field. We
found that transitions between Q_{\theta} subdomain and Q_{\pi/2} and Q_0 ones
occur suddenly but continuously and smoothly, i.e., nonanalyticity is hidden
and can be observed in higher derivatives of discord function.
