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$K^2=1$
We propose two systems of “intrinsic” weights for counting such curves. In both cases the result acquires an exceptionally strong invariance property: it does not depend on the choice of a surface. One of our counts includes all divisor classes of canonical degree 2 and gives in total 30. The other one excludes the class 
$-2K$, but adds up the results of counting for a pair of real structures that differ by Bertini involution. This count gives 96.
