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It is proved that the Heisenberg group $\operatorname*{Nil}\nolimits_{3}$with a balanced metric, the sum of the left and right invariant metrics, splitsas a Riemannian product $\mathbb{T\times Z}$, where $\mathbb{T}$ is a totallygeodesic surface and $\mathbb{Z}$ the center of $\operatorname*{Nil}%\nolimits_{3}.$ It is then proved the existence of complete properly embeddedminimal surfaces in $\operatorname*{Nil}\nolimits_{3}$ by solving theasymptotic Dirichlet problem for the minimal surface equation on $\mathbb{T}$.It is also proved the existence of complete properly embedded minimal surfacesfoliating an open set of $\operatorname*{Nil}\nolimits_{3}$ having as boundarya given curve $\Gamma$ in $\mathbb{T},$ satisfying the exterior circlecondition, by solving the exterior Dirichlet problem for the minimal surfaceequation in the unbounded connected component of $\mathbb{T}\backslash\Gamma$.

Asymptotic and exterior Dirichlet problems for the minimal surface equation in the Heisenberg group with a balanced metric