Over the past two decades much effort was put into the search for dark matter. To the present moment many models were proposed to describe the phenomena, yet there are no observations of the elementary particles that are capable of giving a proper explanation for this phenomenon. This fact gave rise to many attempts to describe dark matter as a purely gravitational effect. Most known model of this kind is Modified Newton Dynamics (MOND), proposed by M. Milgrom in 1981. Though the idea was not only elegant but also accurate in explaining many observations regarding dark matter, it has severe problem in describing the dynamics of the dark matter halo in the Bullet Cluster.

Fairly recently many new novel MOND-like models were proposed, that overcomes this problem. Most notable example is mimetic gravity proposed by A. Chamseddine and V. Mukhanov in 2014. The key idea is to use degenerate conformal transformation in Einstein-Hilbert action to isolate conformal mode of metric into new scalar field:
$$
g_{\mu\nu} = \bar{g}_{\mu\nu}\bar{g}^{\alpha\beta}\partial_\alpha\varphi\partial_\beta\varphi,
$$
where $\bar{g}_{\alpha\beta}$ is auxiliary metric. Because the scalar field enters the change through derivatives, the equations of motion are no longer Einstein but have a new terms that can be treated as effective energy-momentum tensor for dark matter. In the original formulation this matter is just pressureless dust, however its $4$-velocity must be gradient of the scalar, i.e. it is moving potentially. Most importantly, this type of effective dark matter has its own equations of motion and can exist on its own bypassing MOND problems with Bullet Cluster.

The original mimetic gravity and its numerous modifications were extensively studied during the last few years. Most of them rely on the addition of potential to the Lagrange multiplier formulation of the original theory or on the introduction new fields with possibly higher derivatives. In the present talk we are interested in one such modification - $p$-form mimetic gravity. Previously it was shown that it can be dualized in the same manner as the usual $p$-form electrodynamics. Dual theory appears to describe the fluid consisting of $(p+1)$-branes embedded in the spacetime. In the present report we will cover several exact solutions for these models in case of $d = 4$ such as FLRW universes and spherically symmetric cases. Using this formalism we recover the exact solutions $p = 1$ - mimetic gravity, that were previously obtained in the literature and also present the results for the new case $p = 2$, that was thought equivalent to $p = 0$. Finally, we discuss the stability of the obtained solutions are their cosmological implications.