Narrow Lie Algebras and Integrable Complex Structures

We study the existence of an integrable complex structure on a real finite-dimensional naturally graded Lie algebra that is narrow in the sense of Zelmanov and Shalev. Every such Lie algebra is generated by two elements. As the most elementary examples, one can consider naturally graded filiform Lie algebras. We show that almost all such nilpotent Lie algebras admit no integrable complex structures. The only exceptions for which the question under study has a positive answer are the even-dimensional quotient Lie algebras $\mathfrak n_1^+(s)$ obtained by factoring the infinite-dimensional subalgebra $\mathfrak n_1^+$ of the loop algebra $\mathcal L(\mathfrak {so}(3))$ of the real simple Lie algebra $\mathfrak {so}(3)$ by the ideals $(\mathfrak n_1^+)^{s+1}$ of the lower central series.