  In this work, we study the existence of asymptotically Lifshitz black holes
in Critical Gravity in four dimensions with a negative cosmological constant
under two scenarios: First, including dilatonic fields as the matter source,
where we find an asymptotically Lifshitz solution for a fixed value of the
dynamical exponent $z=4$. As a second case, we also added a non-minimally
coupled scalar field $\Phi$ with a potential given by a mass term and a quartic
term. Using this approach, we found a solution for $z$ defined in the interval
$(1,4)$, recovering the Schwarzchild-Anti-de Sitter case with planar base
manifold in the isotropic limit. Moreover, when we analyzed the limiting case
$z=4$, we found that there exists an additional solution that can be
interpreted as a stealth configuration in which the stealth field is overflying
the $z=4$ solution without the non-minimally coupled field $\Phi$. Finally, we
studied the non-trivial thermodynamics of these new anisotropic solutions and
found that they all satisfy the First Law of Thermodynamics as well as the
Smarr relation. We were also able to determine that the non-stealth
configuration is thermodynamically preferred in this case.
