  The simplest and the most straightforward new algorithm for generating
solutions to (anti) self-dual Yang-Mills (YM) equation in the typical
gravitational instanton backgrounds is proposed. When applied to the Taub-NUT
and the Eguchi-Hanson metrics, the two best-known gravitational instantons, the
solutions turn out to be the rather exotic type of instanton configurations
carrying finite YM action but generally fractional topological charge values.
