computing killing vectors

The following system of 15 first order PDEs eq1=0,..,eqF=0 computes all
Killing vectors of a certain 5-dimensional space. It is a system for the 5
functions X1,..,X5 of the 5 variables f, chi, phi, v, u.  It is slightly
complicated through the fact that also a constant `a' is to be determined such
that solutions exist.

eq1 := X1 - df(X1,f)*f;
eq2 := df(X2,f) - u*df(X4,f) + v*df(X5,f) + df(X1,chi);
eq3 := 4*df(X3,f)*f**2 + df(X1,phi);
eq4 := u*df(X2,f) - df(X4,f)*u**2 + 2*df(X4,f)*exp(-2*a*phi)*f + u*v*df(X5,f)
- df(X1,v);
eq5 := v*df(X2,f) - u*v*df(X4,f) + df(X5,f)*v**2 - 2*df(X5,f)*exp(2*a*phi)*f +
df(X1,u);
eq6 := f*df(X2,chi) - f*u*df(X4,chi) + f*v*df(X5,chi) - X1;
eq7 := 4*df(X3,chi)*f**2 + df(X2,phi) - u*df(X4,phi) + v*df(X5,phi);
eq8 := f*u*df(X2,chi) - f*df(X4,chi)*u**2 + 2*f**2*df(X4,chi)*exp(-2*a*phi) +
f*u*v*df(X5,chi) - 2*X1*u + X5*f - f*df(X2,v) + f*u*df(X4,v) - f*v*df(X5,v);
eq9 := f*v*df(X2,chi) - f*u*v*df(X4,chi) + f*df(X5,chi)*v**2 -
2*f**2*df(X5,chi)*exp(2*a*phi) - 2*X1*v + X4*f + f*df(X2,u) - f*u*df(X4,u) +
f*v*df(X5,u);
eqA := df(X3,phi);
eqB := u*df(X2,phi) - df(X4,phi)*u**2 + 2*df(X4,phi)*exp(-2*a*phi)*f +
u*v*df(X5,phi) - 4*df(X3,v)*f**2;
eqC := v*df(X2,phi) - u*v*df(X4,phi) + df(X5,phi)*v**2 -
2*df(X5,phi)*exp(2*a*phi)*f + 4*df(X3,u)*f**2;
eqD := -f*u*df(X2,v) + f*df(X4,v)*u**2 - 2*f**2*df(X4,v)*exp(-2*a*phi) +
f*u*X5 - f*u*v*df(X5,v) + X1*f*exp(-2*a*phi) - X1*u**2 +
2*X3*exp(-2*a*phi)*a*f^2;
eqE := f*v*df(X2,v) - f*u*X4 - f*u*v*df(X4,v) + f*df(X5,v)*v**2 - f*v*X5 -
2*f**2*exp(2*a*phi)*df(X5,v) + 2*X1*u*v - f*u*df(X2,u) + f*df(X4,u)*u**2 -
2*f**2*exp(-2*a*phi)*df(X4,u) - f*u*v*df(X5,u);
eqF := -f*v*df(X2,u) - f*v*X4 + f*u*v*df(X4,u) - f*df(X5,u)*v**2 +
2*f**2*df(X5,u)*exp(2*a*phi) - X1*exp(2*a*phi)*f + X1*v**2 +
2*X3*a*f**2*exp(2*a*phi);

The system is solved by CRACK on a 3GHz AMD64 PC under Linux in 
less than 5 sec
but the user who posted that problem to me claims that he tried different
software for computing differential Groebner Bases which gave different
number of solutions which would make it interesting to run this system
for comparison with rif and diffalg.