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The monodromy transform and corresponding integral equation method described here give rise to a general systematic approach for solving integrable reductions of field equations for gravity coupled bosonic dynamics in string gravity, supergravity and pure vacuum gravity in four and higher dimensions. For string gravity in space-times of $D\ge 4$ dimensions with $d=D-2$ commuting isometries and any number $n$ of Abelian vector gauge fields the equivalent spectral problem allows to parameterize the infinite-dimensional space of local solutions by two pairs of \cal{arbitrary} coordinate-independent holomorphic $d\times d$- and $d\times n$- matrix functions $\{\mathbf{u}_\pm(w),\, \mathbf{v}_\pm(w)\}$ of the spectral parameter $w$ -- the monodromy data for the fundamental solution of our spectral problem. We construct the linear singular integral equations which determine the solutions for any choice of these monodromy data. For any \emph{rational} and \emph{analytically matched} ($\mathbf{u}_+\equiv\mathbf{u}_-$ and $\mathbf{v}_+ \equiv \mathbf{v}_-$) monodromy data the solutions can be found explicitly. Simple reductions of the space of monodromy data lead to solutions for $5D$ minimal supergravity and vacuum gravity in $D\ge 4$ dimensions.

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