Given an arbitrary first-order Néel-type Skyrmion on a spatially varying depolarizing media X , we define the Stokes field h as follows

for some d 1 , d 2 , d 3 satisfying the same properties as in the type-I case, that is $\epsilon >0$ everywhere and $$0 < {(1-t){d_{i}}({p})+t} {0} \ ({i=1}{2}{3} \ )$$ for all $${p}\in {X}$$ and t ∈ [ 0 , 1 ], and the modified $$(\varphi -\pi/2, \lambda ))$$ parametrization $${{\mathcal{S}}}{i}\left({p}^{{}^{{}}}\right)=f\left(\left((1-t){d{i}}\left({p}^{{{}^{}}{{}}}t)\kern0.5em {+} \kern0.5em t\right. {\displaystyle }{\tilde{\mathcal{S}}}i\left ({p}^{\left(\raisebox{-0.8pt}{\tiny$\displaystyle \sim $}\kern-0.2ex \lower-0.7pt\hbox{$+$}} \right)}\right),\left((1-t){d{j}}({p} ^{t})+\kern0.5em t\right) {\tilde{S}}j(p,t)$$ for i, j = 1, 2, 3 , where $$f(p{i},p_{2})(s)=\frac{\left|\begin{s}{i}\kern-0.5em \hat{i}& s{j}\kern-0.5em \hat{j}\right|}{\sqrt{{p}{i}^{{}^{2}}+{p}{{}$ ^{{}2}^{} }}},$$ where $$s_{1},{s}{2},{s}{3}({p})={(f_{1}(p)\kern-0.5em \hat{i}+ f_{2}(p)\kern-0.5em \hat{j}+ f_{3}(p)\kern-0.5em \hat{k}}\right). ^{\mathrm{H}}$ $ is a continuous homotopy $X\times [0,1]\longrightarrow S^{2}$ given by