The noncompact toric variety $X_1 = \operatorname{Tot} \mathcal{O}(-1) + \mathcal{O}(-1) \to \mathbb{CP}^1$, the total space of the sum of two line bundles over the complex line, is defined as the quotient $X = \mathbb C^4_{ss}/\mathbb C^\star$, where $\mathbb C^\star$ acts on $x=(x_1,\ldots,x_4) \in \mathbb C^4$ with charges $(-1,-1,1,1)$, namely $(x_1,x_2,x_3,x_4) \mapsto (t^{-1}x_1,t^{-1}x_2,t x_3,t x_4)$, and we defined $\mathbb C^4_{ss} = \mathbb C^2 \times (\mathbb C^2 \setminus (0,0))$. Such $X$ can also be represented as a symplectic quotient, in the following sense: let momenta $p_i = |x_i|^2$ and define $\mu_1(x) := -p_1 - p_2 + p_3 + p_4$. Then $X= \mu_1^{-1}(t)/U(1)$ for any real $t>0$ and with $U(1)$ acting as the real part of $\mathbb C^\star$, namely $(x_1,x_2,x_3,x_4) \mapsto (e^{-\sqrt{-1}\alpha}x_1,e^{-\sqrt{-1}\alpha}x_2,e^{\sqrt{-1}\alpha} x_3,e^{\sqrt{-1}\alpha} x_4)$.
Now consider the toric variety $X_2 = \operatorname{Tot} \mathcal{O}(-4) + \mathcal{O}(2) \to \mathbb{CP}^1$, given by the quotient of the same $\mathbb C^4_{ss}$ by the $\mathbb C^*$ acting as $(x_1,x_2,x_3,x_4) \mapsto (t^{2}x_1,t^{-4}x_2,t x_3,t x_4)$. Such $X_2$ is also non-compact, but it is non-nef. Can it be represented as a symplectic quotient, for some choice of moment map? Clearly $\mu_2(x) = 2p_1-4p_2+p_3+p_4$ does not work.
A reference I was reading is Lerman's paper on symplectic cuts. Eugene Lerman: Symplectic cuts, Mathematical Research Letters 2 (1995), 247–258
