= Solution
{c}
The canted pattern has opposite $x$ components but a common $z$ component. Write neighboring spins as $\mathbf S_A=S(\cos\theta,0,\sin\theta)$ and $\mathbf S_B=S(-\cos\theta,0,\sin\theta)$. Their scalar product is $-S^2\cos2\theta$, and the cross-product magnitude is $S^2|\sin2\theta|$.
There are three exchange bonds per spin, two eligible $xy$ bonds per spin for the $D$ term, and one single-ion term per spin. Hence
$$
E=-S^2[3J\cos2\theta+K\cos^2\theta+2D|\sin2\theta|].
$$
For the branch $\theta\ge0$ close to the collinear state,
$$
\boxed{f(\theta)=\cos2\theta,\qquad g(\theta)=\cos^2\theta,\qquad h(\theta)=\sin2\theta.}
$$
Minimization is equivalent to maximizing $(3J+K/2)\cos2\theta+2D\sin2\theta$, which gives
$$
\boxed{\tan2\theta=\frac{4D}{6J+K},\qquad\theta=\frac12\arctan\frac{4D}{6J+K}\simeq0.00518\,\mathrm{rad}\simeq0.297^\circ.}
$$
The chosen branch is a minimum of the energy because it maximizes that positive-amplitude sinusoid. The resulting state is a <canted antiferromagnet> with a small common $z$ moment. The source's absolute-value interaction selects a canting magnitude but is symmetric between the two signs of the weak moment; it is not an oriented vector <Dzyaloshinskii-Moriya interaction> specifying a unique handedness.
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