  The well known potential BKP hierarchy and the dispersionless potential BKP
hierarchy are rewritten via mastersymmetry approach. For the potential BKP
hierarchy, a decomposition conjecture is proposed: there exist six possible
different types of decompositions for every order potential BKP equation
including KdV and higher order KdV decompositions. These decompositions allow
another conjecture about linear superpositions, namely, only two of the
decomposition solutions admit three special types of linear superposition
solutions for every equation of the potential BKP hierarchy. The correctness of
the decompositions and the linear superposition conjectures is clearly verified
for the fifth, seventh and ninth order potential BKP equations. The linear
superposition solutions provide us with a comprehensive approach to construct
many new exact solutions. Particularly, several illustrative examples of exact
linear superposition solutions of the fifth potential BKP equation are given.
In addition, similar decompositions and linear superposition phenomena are also
studied for the dispersionless potential BKP hierarchy. Interestingly, the
possible linear superposition ways increase rapidly as the order of the
dispersionless potential BKP equation arises.
