In recent years KdV-type hierarchies have been related to 2D gravity. The square root of the partition function of the hermitian \((n-1)\)-matrix model in the continuum limit is the \(\tau\)-function of the \(n\)-reduced KP hierarchy. The partition function is characterized by the string equation \(L_{-1}\tau=0\), where \(L_{-1}\) is an element of the Virasoro algebra naturally associated with the principal realization of the affine Lie algebra \(A^{(1)}_{n-1}\). It is known that the \(n\)-reduced KP hierarchy together with the string equation imply more general constraints for a \(\tau\)-function, namely the vacuum constraints of the \(W_{1+\infty}\) algebra, the central extension of the Lie algebra of differential operators on \(\mathbb{C}^x\).
In the paper the author studies an analogous problem for the \(n\)-reduced BKP hierarchy, where \(n\) is assumed to be odd. This reduction is related to the principal realization of the basic module over the affine Lie algebra \(A^{(2)}_{n-1}\). The main theorem is as follows. If one poses the constraint that the \(\tau\)-function satisfies the string equation \(L_{-1}\tau=0\) for the naturally attached Virasoro algebra, then \(\tau\) also satisfies the vacuum constraints of the so-called \(BW_{1+\infty}\) algebra, which is described as a subalgebra of \(W_{1+\infty}\) consisting of the anti-symmetric elements under an anti-involution on \(W_{1+\infty}\).