  We present the method of group foliation for constructing non-invariant
solutions of partial differential equations on an important example of the
Boyer-Finley equation from the theory of gravitational instantons. We show that
the commutativity constraint for a pair of invariant differential operators
leads to a set of its non-invariant solutions. In the second part of the paper
we demonstrate how the hodograph transformation of the ultra-hyperbolic version
of Boyer-Finley equation in an obvious way leads to its non-invariant solution
obtained recently by Manas and Alonso. Due to extra symmetries, this solution
is conditionally invariant, unlike non-invariant solutions obtained previously.
We make the hodograph transformation of the group foliation structure and
derive three invariant relations valid for the hodograph solution, additional
to resolving equations, in an attempt to obtain the orbit of this solution.
