Let $X$ be a smooth manifold of dimension $n$, let $\pi:T^*X\setminus 0 \to X$ be the cotangent projection, and let $\Lambda \subset T^*X\setminus 0$ be a smooth conic Lagrangian submanifold with respect to the canonical symplectic form. For every $\lambda_0=(x_0,\xi_0) \in \Lambda$, with $x_0 := \pi(\lambda_0)$ and $\xi_0\neq 0$, there exist an integer $N$ with $1 \leq N \leq n$, an integer $k=n-N$, a coordinate neighbourhood $U \subset X$ of $x_0$ with local coordinates $x=(x',x'')\in\mathbb{R}^k\times\mathbb{R}^N$, a conic [open set](/page/Open%20Set) $\Theta \subset \mathbb{R}^N_0$, and a smooth function $\phi:U \times \Theta \to \mathbb{R}$ that is homogeneous of degree one in $\theta \in \Theta$ and nondegenerate in the sense that the differentials $d_{x,\theta}(\partial_{\theta_j}\phi)$, $1 \leq j \leq N$, are linearly independent on $C_\phi := \{(x,\theta) \in U \times \Theta : \partial_\theta \phi(x,\theta)=0\}$. There is also an open conic neighbourhood $W \subset T^*U\setminus 0$ of $\lambda_0$ such that $\Lambda \cap W = \Lambda_\phi \cap W$, where $\Lambda_\phi := \{(x,\partial_x\phi(x,\theta)) : (x,\theta)\in C_\phi\}$. Moreover, there exist a point $(x_0,\theta_0)\in C_\phi$ satisfying $j_\phi(x_0,\theta_0)=\lambda_0$ and a conic open neighbourhood $\Omega\subset U\times\Theta$ of $(x_0,\theta_0)$ such that, with $C_\phi^{\mathrm{loc}}:=C_\phi\cap\Omega$ and $j_\phi:C_\phi \to T^*U\setminus 0$ defined by $j_\phi(x,\theta)=(x,\partial_x\phi(x,\theta))$, the restriction $j_\phi|_{C_\phi^{\mathrm{loc}}}:C_\phi^{\mathrm{loc}}\to \Lambda\cap W$ is a diffeomorphism.