In a complete general framework, probably as you know, you can define an hamiltonian system as a dynamical system whose vector field reads: $$[X_{H}(x)]{i}= {x{i},H} $$ where $$x=(q_{1},...,q_{n},p_{1},...,p_{n})=(q,p)$$ is a vector of the phase space $\Gamma$ (that is practically always an Hilbert space), ${ ; }$ is a Poisson Bracket that satisfies the usual properties and $H=H(x)$ is the Hamiltonian of the system, defined on the phase space. A really important object that one can define, is the so-called Poisson tensor, defined as: $$J_{jk} = {x_{j},x_{k}}$$ Note that from this definition is obvious that $J$ has got the same skew-symmetry property of the <PERSON> brackets: this is fundamental in order to redefine a formulation of any Hamiltonian system in terms of $J$. At this point, without enter a lot in details, one can develope what is written above redefining the <PERSON> brackets in terms of the Poisson tensor and in this way one can write a complete general form of the <PERSON> equation of ANY Hamiltonian system: $$\dot{x} =J(x)\nabla_{x}H(x)$$ where $J$ is the Poisson tensor and the notation $\nabla_{x}$ means that the gradient acts on a vector of the phase space $x$. You can make ANY change of variable $x \rightarrow y=f(x)$ that you want, but you will NEVER lose the hamiltonian properties of your system (i.e. the properties of the Poisson brakets). 976
This is really important and means that if a system is Hamiltonian, according to the definitions above, it will remains Hamiltonian INDIPENDENTLY of the coordinate chosen to describe it.
The concept of the Poisson tensor is also important in order to characterize the set of Casimir invariants (i.e the symmetry group) of a given Hamiltonian system: A Casimir invariant $C(x)$ is a function defined on the phase space, such that: $$J(x)\nabla C(x)=0$$ in other words, is a function which has got the gradient that is in the Kernel of the Poisson tensor. These functions describe the invariace of your system. The main point (that i hope can answers to your question) is easy to understand just from the definition of a <PERSON> invariant and is that ANY Hamiltonian system that is described by the same Poisson tensor, has got the same <PERSON> invariants (i.e the same symmetry)! Is not correct the way in which your are posing your question, beacause every Hamiltonian system has got a well defined algebra of functions $A(\Gamma)$ whose bilinear product is a <PERSON> bracket that satisfies the usual properties that you know: if is not possible to define a <PERSON> bracket you can't have any Hamiltonian system. It's not meaningful what you are asking for beacause the definition of the classical Hamiltonian algebra is really general and , i repeat, well-defined in any case. In other words, the Hamiltonian algebra defined on the phase space doesn't count if you are talking about invariance and symmetries of an Hamiltonian system but, as i wrote to you above, the main instrument in order to characterize this fact is the Poisson tensor of the system: the same Poisson tensor -> the same Casimir(symmetries) 976