manifold $\mathcal{N}$ is defined by the system of independent equations $$\label{eq4_8}
  f_1=0, \qquad f_2=0.$$ Then the 2-form on ${\mathcal{M}}$ induced by the symplectic structure of $\mathcal{N}$ degenerates exactly on the set $\{f_1,f_2\}=0$.

Since critical subsystems are usually described by the systems of the type , the induced symplectic structure may degenerate on a set of codimension 1. In this case the subsystem is said to be almost Hamiltonian.

The next statement follows from the results of [@RyabTMF] with the above agreement on the Poisson brackets.

\[prop2\] The restriction to ${\mathcal{M}}_1$ of the Poisson bracket $\{\xi_1,\xi_2\}$ is a partial integral of the induced system $$\label{eq4_9}
  \{\xi_1,\xi_2\}= -2 \,N \, F,$$ where $N$ and $F$ are both partial integrals and can be presented as follows $$\label{eq4_10}
   N = \sqrt{\bigl[ {\varepsilon}_0 + \rmi {\varepsilon}_1  (w_3+\rmi {\varepsilon}_1  y_1) \bigr] \bigl[ {\varepsilon}_0 - \rmi {\varepsilon}_1  (w_3-\rmi {\varepsilon}_1  y_2) \bigr]}+{\varepsilon}_1 ^2 \sqrt{\mathstrut x_1 x_2 },$$ $$\label{eq4_11}
   F = \sqrt{\mathstrut x_1 x_2 } w_3-\frac{x_2 z_1 w_1+x_1 z_2 w_2}{\sqrt{\mathstrut x_1 x_2 }}+\rmi {\varepsilon}_1  r^2\frac{x_1-x_2}{\sqrt{\mathstrut x_1 x_2 }}.$$ The connection of the general integrals $H,G$ on ${\mathcal{M}}_1$ with these new integrals is $$p^2 H -2 G+{\varepsilon}_1 ^2 r^4 = \frac{F^2}{2}, \qquad {\varepsilon}_0 ^2+2 {\varepsilon}_1 ^2(H+{\varepsilon}_1 ^2 p^2) = N^2.$$ 