Let the (non-canonical) Hamiltonian system be given in the form $$\dot{x}=J(x)DH(x)$$ where $H(x)$ is the Hamiltonian function, $J(x)$ is the skew-symmetric symplectic matrix, and $DH$ denotes the vector of partial derivatives of $H$ (the gradient of $H$). Let $J(x)$ have a nonempty null space, and let a function $C(x)$ satisfy $J(x)DC(x) = 0$ for all $x$. Show that value of $C(x)$ is constant on any trajectory of the system regardless of the choice of $H$.
Here is my proposed solution. I use the fact that $J(x)$ is the skew-symmetric in the fourth equality $\big(J(x)^{-1} = J(x)^T = -J(x)\big)$. The fact that $J(x)DC(x) = 0$ is used in the fifth equality.
\begin{equation} \begin{split} C\dot{(x)} & = \nabla{C}\boldsymbol{\cdot}{x} \\ & = \nabla{C}\boldsymbol{\cdot}J(x)DH(x) \\ & = \Big((J(x))^T(\nabla{C})^T\Big)^T\boldsymbol{\cdot}DH(x) \\ & = \Big(-J(x)DC(x)\Big)^T\boldsymbol{\cdot}DH(x) \\ & = 0\boldsymbol{\cdot}DH(x) \\ & = 0 \end{split} \end{equation}
Therefore, as $C\dot{(x)}=0$, this implies that $C(x)$ is constant along any trajectory.
I'm having trouble justifying the first equality and fourth equality. Is this the right approach to this problem? Is there an alternative approach that would lead to the right answer?
I think you are thinking on the right line. I have the following solution.
Write, along the solution trajectories of the system
$\begin{align} \frac{d}{dt} H(x(t)) &= \frac{\partial H}{\partial x} \dot{x} \nonumber \\ &= (DH(x))^T J(x) DH(x) \qquad (1)\\ &= (DH(x))^T (J(x))^T DH(x) \quad \\ &= -(DH(x))^T J(x) DH(x) \quad (2)\\ \text{Adding equations (1) and (2), we get,}\\ 2 \frac{d}{dt} H(x(t)) &= 0\\ \Rightarrow \frac{d}{dt} H(x(t)) &= 0 \end{align}$
Hence, you can write $H(x(t)) = c$, for a constant $c$. So, $H(x)$ is constant along the trajectory of the system.