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We study the spectral measures of conservative mixing flows on the 2-torushaving one degenerate singularity. We show that, for a sufficiently strongsingularity, the spectrum of these flows is typically Lebesgue with infinitemultiplicity. For this, we use two main ingredients: 1) a proof of absolutecontinuity of the maximal spectral type for this class of non-uniformlystretching flows that have an irregular decay of correlations, 2) a geometriccriterion that yields infinite Lebesgue multiplicity of the spectrum and thatis well adapted to rapidly mixing flows.

Lebesgue spectrum of countable multiplicity for conservative flows on the torus