  This work is devoted to the studies of a Hamilton--Jacobi equation with a
quadratic and degenerate Hamiltonian, which comes from the dynamics of a
multipeakon in the Camassa--Holm equation. It is given by a quadratic form with
a singular positive semi-definite matrix. We increase the regularity of the
value function considered in our previous paper, which is known to be the
viscosity solution. We prove that for a two-peakon Hamiltonian such solutions
are actually $1/2$-H\"{o}lder continuous in space and time-Lipschitz
continuous. The time-Lipschitz regularity is proven in any dimension $N\geq 1$.
Such a regularity is already known in the one-dimensional simplifications,
moreover it is the best possible, as was shown in our previous papers.
