  We prove the existence of nontrivial closed surfaces with constant
anisotropic mean curvature with respect to elliptic integrands in closed smooth
$3$-dimensional Riemannian manifolds. The constructed min-max surfaces are
smooth with at most one singular point. The constant anisotropic mean curvature
can be fixed to be any real number. In particular, we partially solve a
conjecture of Allard [Invent. Math.,1983] in dimension $3$.
