Math system with external parameter

I’m writing a document where I need to define a system with an external parameter. Here an image to make it clearer:

I’m using this current code to generate a common system, here is an example:

documentclass{report}
usepackage{mathtools}
begin{document}

begin{spreadlines}{1.5ex}
      begin{equation}
        begin{dcases}
        mathbf{dot r} ,= frac{dr}{dt} ;= mathbf{v} 
        mathbf{dot v} ,= frac{dv}{dt} ;= g + frac{mathbf{T}}{m} ;= g + mathbf{a_T}
        dot m = frac{dm}{dt} = -frac{T}{g_o I_mathrm{sp}} = -frac{T}{c}
        end{dcases}
       end{equation}
end{spreadlines}

end{document}

How could I rework the code to make an external parameter appear (likewise the psi greek letter along with the equality sign)?

Thank you!

documentclass{report}
usepackage{mathtools}
begin{document}

begin{spreadlines}{1.5ex}
begin{equation}
psi=
left{
    begin{alignedat}{3}
      dot{mathbf{r}} &= frac{dr}{dt} &&= mathbf{v} 
      dot{mathbf{v}} &= frac{dv}{dt} &&= g + frac{mathbf{T}}{m}     &&= g + mathbf{a_T}
      dot{m}          &= frac{dm}{dt} &&= -frac{T}{g_o I_mathrm{sp}} &&= -frac{T}{c}
    end{alignedat}
right.
end{equation}
end{spreadlines}

end{document}