A Riemannian metric g on a manifold \(M^{4k}\) is said to be quaternionic if its holonomy group is reduced to Sp(1)\(\times Sp(k)\) and is said to be hyper-Kähler if this group is reduced to Sp(k) [S. M. Salamon, Invent. Math. 67, 143-171 (1982; Zbl 0486.53048)]. The last condition is equivalent to the condition that there exists a family of complex structures on M that is parametrized by the points of the projective line CP 1 (with certain natural conditions), with respect to each of which the metric is Kähler. Therefore, it is natural to perceive the hyper- Kählerness as the quaternionic Kählerness.
For \(k=1\) (dim M\(=4)\) the notion of a hyper-Kähler metric transforms into the notion of a right-planar metric: g is the autodual solution of the (vacuum) Einstein equation. The construction of explicit examples of hyper-Kähler metrics is a difficult problem even for \(k=1\). A few examples are known for \(k>1\). Hitchin, Rocek, etc. have announced a general method for the construction of examples that is based on the fact that under natural restrictions the hyper-Kählerness is preserved under factorization with respect to the invariant action of a compact Lie group. Here we propose another method for the construction of hyper- Kähler metrics that generalizes the method of the author [Yad. Fiz. 36, No.2, 537-548 (1982; Zbl 0588.53049), Funct. Anal. Appl. 18, 108-119 (1984); translation from Funkts. Anal. Prilozh. 18, No.2, 26-39 (1984; Zbl 0574.32003), and Funct. Anal. Appl. 19, 210-213 (1985); translation from Funkts. Anal. Prilozh. 19, No.3, 58-60 (1985; Zbl 0599.53050)] for the construction of the right-planar metrics. As there, we regard the hyper-Kählerian structures in a more general class of geometric structures and obtain hyper-Kähler metrics, restricting these more general structures to suitable submanifolds.