  In this article we investigate the stability of quantized yrast (QY) states
in a mixture of two distinguishable equal mass bosonic atoms, $A$ and $B$,
confined in a ring. We focus our investigation in the study of the energetic
stability since the Bloch analysis and the Bogoliubov theory establish that
only energetically stable QY states are capable of sustain a persistent
current. Based on physical considerations the stability is studied in two
different two-dimensional planes. One is when we are studying the stability of
a single QY state which is realized in the $U_{AB}\times U$ plane spanned by
the inter and intraspecies interaction strengths with fixed values of angular
momentum per particle $l$ and population imbalance $f$ equal to the labels of
the QY state. We found that the energetic phase boundary is the positive branch
of a hyperbola and the energetically stable domain the internal region of this
positive branch. The other is when we are studying the stability at a fixed
dynamics which is realized in the $l\times f$ plane spanned by $l$ and $f$ with
fixed values of the interaction strengths. The QY states are introduced when we
postulate a correspondence between points in sector of the $l\times f$ plane of
physical significance (SPS), defined by $-\infty< l<\infty$ and $-1\leq f\leq
1$, and QY states. The stability diagram in the SPS is determined by the
overlap of the stability diagram in all $l\times f$ plane and the SPS. We found
that there are critical values of $f$ and $l$. $f_\mathrm{crit}(l)$ gives the
size of the window of energetic stability in the sense that for a given $l$
only QY states with $0\leq f<f_\mathrm{crit}(l)$ are energetically stable. On
the other hand, $l_\mathrm{crit}$ states that there is none energetically
stable QY state with $l>l_\mathrm{crit}$.
