Let $ K $ be a field, $ char K = 0, $ let $ Q = x_1^2+\dots+x_n^2-t_1^2-\dots-t_n^2 $ be the totally isotropic form, then the maximal linear isotropic subspaces of $ V(Q) \subset K^{2n} $ are the translates of $ V(x_1-t_1,\dots,x_n-t_n) $ under the action of $ O(n,n) $ and therefore have dimension $ n. $

Question: Do there exist isotropic homogeneous subvarieties of $ V(Q) $ of dimension $ n $ which are not linear subspaces ?

Let $ K $ be a field, $ \operatorname{char} K = 0, $ let $ Q = x_1^2+\dots+x_n^2-t_1^2-\dots-t_n^2 $ be the totally isotropic form, then the maximal linear isotropic subspaces of $ V(Q) \subset K^{2n} $ are the translates of $ V(x_1-t_1,\dots,x_n-t_n) $ under the action of $ O(n,n) $ and therefore have dimension $ n. $

Question: Do there exist isotropic homogeneous subvarieties of $ V(Q) $ of dimension $ n $ which are not linear subspaces ?