$d\le 1\cup d\ge 25$ and constrained KP hierarchy from BRST invariance in the $c\ne 3$ topological algebra

The BRST invariance condition in a highest-weight representation of the topological ($\equiv$ twisted $N=2$) algebra captures the ‘invariant’ content of two-dimensional gravity coupled to matter.The topological algebra allows reductions to either the DDK-dressed matter or the ‘Kontsevich-Miwa’-dressed matter related to Virasoro-constrained KP hierarchy. The standard DDK formulation is recovered by splitting the topological generators into $c=-26$ reparametrization ghosts + matter + ‘Liouville’, while a similar splitting involving $c=-2$ ghosts gives rise to the matter dressed in exactly the way required in order that the theory be equivalent to Virasoro constraints on the KP hierarchy. The two dressings of matter with the ‘Liouville’ differ also by their ‘ghost numbers’, which is similar to the existence of representatives of BRST cohomologies with different ghost numbers. The topological central $c\ne 3$ provides a two-fold covering of the allowed region $d\le 1\cup d\ge 25$ of the matter central charge $d$ via $d+(c+1)(c+6)(c-3)$. The ‘Liouville’ field is identified as the ghost-free part of the topological $U(1)$ current. The construction thus allows one to establish a direct relation (presumably an equivalence) between the Virasoro-constrained KP hierarchies, minimal models, and the BRST invariance condition for highest-weight states of the topological algebra.