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The reciprocal link between the reduced Ostrovsky equation and the$A_2^{(2)}$ two-dimensional Toda system is used to construct the $N$-solitonsolution of the reduced Ostrovsky equation. The $N$-soliton solution of thereduced Ostrovsky equation is presented in the form of pfaffian through ahodograph (reciprocal) transformation. The bilinear equations and the$\tau$-function of the reduced Ostrovsky equation are obtained from the period3-reduction of the $B_{\infty}$ or $C_{\infty}$ two-dimensional Toda system,i.e., the $A_2^{(2)}$ two-dimensional Toda system. One of $\tau$-functions ofthe $A_2^{(2)}$ two-dimensional Toda system becomes the square of a pfaffianwhich also become a solution of the reduced Ostrovsky equation. There isanother bilinear equation which is a member of the 3-reduced extended BKPhierarchy. Using this bilinear equation, we can also construct the samepfaffian solution.

On the $\tau$-functions of the reduced Ostrovsky equation and the $A_2^{(2)}$ two-dimensional Toda system