We study invariant manifolds of conformal symplectic dynamical systems on a symplectic manifold ( ℳ , ω ) of dimension ≥ 4 . We first prove the ω -isotropy of an invariant manifold 𝒩 , assuming the entropy of 𝒩 is small with respect to the conformality rate. Next, when ( ℳ , ω ) is exact and 𝒩 is isotropic, we show that 𝒩 must be exact for some choice of the primitive of ω , under the condition that the dynamics acts trivially on the cohomology of degree 1 of 𝒩 . The conclusion partially extends if a one-sided orbit of 𝒩 has compact closure. We eventually describe some conditions showing the uniqueness of 𝒩 .