  Recently, Doroudiani and Karimipour [Phys. Rev. A \textbf{102} 012427(2020)]
proposed the notation of planar maximally entangled (PME) states which are a
wider class of multipartite entangled states than absolutely maximally
entangled (AME) states. There they presented their constructions in the
multipartite systems but the number of particles is restricted to be even. Here
we first solve the remaining cases, i.e., constructions of planar maximally
entangled states on systems with odd number of particles. In addition, we
generalized the PME to the planar $k$-uniform states whose reductions to any
adjacent $k$ parties along a circle of $N$ parties are maximally mixed. We
presented a method to construct sets of planar $k$-uniform states which have
minimal support.
