General solution of the SU(infinity)-Toda equation
Determine the general solution of the two-dimensional or three-dimensional SU(infinity)-Toda equation arising in the LeBrun–Tod ansatz for one-sided type-D Ricci-flat metrics.
References
The $SU(\infty)$-Toda equation (ref{Toda_3D}) is non-linear, and its general solution is not known.
A direct interpretation of this potential is not yet known.
With the observation that solutions in even columns in Table~\ref{table_multisoliton} are ALF and those in odd columns are ALE, we are led to make the following conjecture:
We can prove that, in each of (\ref{club_conjecture}) and (\ref{diamond_conjecture}), the rod structures of the solutions on both sides of the symbol $\sim$ match exactly via parameter redefinitions, but we fail to map the ranges of parameters\footnote{The ranges of parameters of multi-soliton solutions on flat space were not given in . It is a problem worth further investigation.} and prove the full equivalence of corresponding metrics.
It is well known that the multi-centre Gibbons--Hawking metrics admit generalisations in which the centres are not all aligned along any axis, and the metrics in this case have only one Killing vector. Do one-sided type-D Ricci-flat multi-centre metrics have generalisations with centres not all aligned on any axis? There is also only one Killing vector, guaranteed by the one-sided type-D condition. The Weyl--Papapetrou coordinates are of course inadequate to describe such metrics (if they exist at all).