Trivial holonomy gives a product foliated neighbourhood

(The holonomy cover is trivial.) With ${\rho }_x$ trivial, the kernel is all of ${\pi }_1(L,x)$, so the covering $p:\widehat{L}\to L$ associated with the kernel has $p_{\ast }{\pi }_1(\widehat{L},\widehat{x})={\pi }_1(L,x)$; a covering of degree one is a diffeomorphism, and we identify $\widehat{L}=L$ [F1].

(The model and the neighbourhood.) With $H=\{1\}$ the finite-holonomy normal model is the product $L\times D$ with the product foliation by slices [F2]. The local Reeb stability theorem applies because the holonomy group is finite (indeed trivial), and gives, for every neighbourhood $W$ of $L$, a saturated neighbourhood $U\subseteq W$ foliated-diffeomorphically onto a neighbourhood of the central leaf, which after shrinking $D$ is a product $L\times D^{\mathrm{\prime }}$ with the product foliation [F2, F3].

(Fundamental system and leaves.) The product neighbourhoods $L\times D^{\mathrm{\prime }}$ for shrinking transverse disks $D^{\mathrm{\prime }}$ form a fundamental system of neighbourhoods of the central leaf, and each leaf of the product foliation is a slice $L\times \{t\}$, compact and diffeomorphic to $L$ [F2, F4]. Hence $L$ has a fundamental system of product foliated neighbourhoods whose leaves are compact and diffeomorphic to $L$.