We present a systematic way to construct solutions of the (n=5)-reduction of
the BKP and CKP hierarchies from the general tau function of the KP hierarchy.
We obtain the one-soliton, two-soliton, and periodic solution for the
bi-directional Sawada-Kotera (bSK), the bi-directional Kaup-Kupershmidt (bKK)
and also the bi-directional Satsuma-Hirota (bSH) equation. Different solutions
such as left- and right-going solitons are classified according to the
symmetries of the 5th roots of exp(i epsilon). Furthermore, we show that the
soliton solutions of the n-reduction of the BKP and CKP hierarchies with n= 2 j
+1, j=1, 2, 3, ..., can propagate along j directions in the 1+1 space-time
domain. Each such direction corresponds to one symmetric distribution of the
nth roots of exp(i epsilon). Based on this classification, we detail the
existence of two-peak solitons of the n-reduction from the Grammian tau
function of the sub-hierarchies BKP and CKP. If n is even, we again find
two-peak solitons. Last, we obtain the "stationary" soliton for the
higher-order KP hierarchy.