  In this paper, we present two new families of spatially homogeneous black
hole solution for $z=4$ Ho\v{r}ava-Lifshitz Gravity equations in $(4+1)$
dimensions with general coupling constant $\lambda$ and the especial case
$\lambda=1$, considering $\beta=-1/3$. The three-dimensional horizons are
considered to have Bianchi types $II$ and $III$ symmetries, and hence the
horizons are modeled on two types of Thurston $3$-geometries, namely the Nil
geometry and $H^2\times R$. Being foliated by compact 3-manifolds, the horizons
are neither spherical, hyperbolic, nor toroidal, and therefore are not of the
previously studied topological black hole solutions in Ho\v{r}ava-Lifshitz
gravity. Using the Hamiltonian formalism, we establish the conventional
thermodynamics of the solutions defining the mass and entropy of the black hole
solutions for several classes of solutions. It turned out that for both horizon
geometries the area term in the entropy receives two non-logarithmic negative
corrections proportional to Ho\v{r}ava-Lifshitz parameters. Also, we show that
choosing some proper set of parameters the solutions can exhibit locally stable
or unstable behavior.
