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Let $\mathfrak{g}=< X_{f_1},X_{f_2},...,X_{f_n}> $ be a lie algebra of contact symmetries , $\mathfrak{g}\subset Sym(E_\omega )$ , such that the orbit spact $ B_\mathfrak{g}=E_\mathfrak{g}/ \mathfrak{g}$ is smooth Then can we say $(B_\mathfrak{g}, C_\mathfrak{g}, \omega_\mathfrak{g})$ is contact manifold? where if $\pi :E_\mathfrak{g}\rightarrow B_\mathfrak{g}$ be the natural projection and $C_\mathfrak{g}([a])=\pi_*(C_\mathfrak{a}\cap T_a E_\mathfrak{g})$ and $[a]$ is orbit of $a\in E_\mathfrak{g}$ and $\omega_{\mathfrak{g}}=i_{X_{f_1}}\circ i_{X_{f_2}}\circ ...i_{X_{f_k}}(\omega)$ .Here $\omega\in \Omega^n(J^1M)$, and by sense of V.Lychagin we denote the monge ampere equation $\Delta_\omega=0$ by $E_\omega$ . Moreover $E_{\mathfrak{g}}$ is the set of solutions of $(f_1=0,f_2=0,....,f_k=0 )$ . .You can find this fact as theorem 12.3.1 link text page 258 In fact in this book the authors used of this fact and never proved it

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 I'm afraid most people aren't going to have time to read through a book to sort out its notation so that they can help you with this question. You'll need to put the definitions of unexplained terms, such as $\omega$, $E_\omega$, and $E_{\frak{g}}$, etc. and explain your notation before we can help you. – Robert Bryant Mar 15 at 17:37
In fact this question is concerning about reduction of monge ampere equations(in sense of Lychagin) on contact manifolds – Hassan Jolany Mar 15 at 18:39
There is a paper of Christopher Willet about contact reduction (arxiv.org/abs/math/0104080), but it doesn't have any theorem in it that I can immediately recognize as solving your problem. I suspect that the Monge-Ampere equation is irrelevant here. Are your contact symmetries preserving a particular choice of contact 1-form, or only a contact structure (with Monge-Ampere equation, obviously)? – Ben McKay Mar 15 at 22:57
In fact the definition of Monge-ampere equation of a contact manifold $(N^{2r+1}, \omega)$, is 1-dimensional subbundle $l$ of the bundle effective r-forms of the distribution $ker \omega$. and in my question $\omega_\mathfrak{g}$ define a monge ampere equation $l_\mathfrak{g}$ and Also "reduction" here , means that the projection of any $\mathfrak{g}$-invariant solution $L$ of $E_\omega$ with smooth orbit space $L/\mathfrak{g}$ is a solution of $l_\mathfrak{g}$ . – Hassan Jolany Mar 16 at 20:45
@ Ben McKay, in fact we say contact vector field $X_f$ on the manifold $J^1M$ is an infinitesimal symmetry of the Monge-ampere equation $E_\omega$ if $L_{X_f}(\omega)=h\omega$ mod $I^n$, where $I^n$ is effective forms and so contact symmetry $\phi\in Ct(J^1M)$ preserve contact 1_form module the set of effective forms – Hassan Jolany Mar 16 at 20:53

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