f := (i,j) -> `if`(j=1, i^2, i^3):
Matrix(10,2,f);

The system is too complicated to be solved. Use fsolve instead, it works.

Do the same for "Cello" (and conclude).

Definitely the error is due to your "..." (not shown).

You may have used it incorrectly, but Groebner solves at once your problem.

sys:=O1*O3*f7-f1, O1*O3*f1-f7, O1*O3*f8-f2, O1*O3*f2-f8, O1*O3*f5-f3, O1*O3*f3-f5, O1*O3*f6-f4, O1*O3*f4-f6, O1*O4*f8-f1, O1*O4*f1-f8, O1*O4*f7-f2, O1*O4*f2-f7, O1*O4*f6-f3, O1*O4*f3-f6, O1*O4*f5-f4, O1*O4*f4-f5, O1*O2*O3*f6-f1, O1*O2*O3*f1-f6, O1*O2*O3*f5-f2, O1*O2*O3*f2-f5, O1*O2*O3*f8-f3, O1*O2*O3*f3-f8, O1*O2*O3*f7-f4, O1*O2*O3*f4-f7, O1*O2*O4*f5-f1, O1*O2*O4*f1-f5, O1*O2*O4*f6-f2, O1*O2*O4*f2-f6, O1*O2*O4*f7-f3, O1*O2*O4*f3-f7, O1*O2*O4*f8-f4, O1*O2*O4*f4-f8:

oo:=O1^2-1,O2^2-1,O3^2-1,O4^2-1:

Groebner[Basis]([sys,oo], plex(f1, f2, f3, f4, f5, f6, f7, f8,  O1, O2, O3, O4)):
select(has,%,[seq(f||i,i=1..8)]);

You may solve eqs wrt all the variables:

eqs := {x - a*y,y - a*x}:
solve(eqs);

n:=3:
Matrix(combinat[permute]( [-1$n, 1$n], n) );

seq(x||i,i=1..200);

So, you have a homogeneous linear system of ODEs with constant coefficients.
If you know a fundamental system for the solutions:

exp(p*t), exp(q*t), ...

where p,q, ... are distinct complex numbers
then the characteristic polynomial in variable y is

(y - p)(y - q) ...

[if there are multiple roots, the situation is a bit more complicated]

Note that c(t) = 0 cannot appear in a fundamental system
(because of the linear independence).

indets(eq,anyfunc(identical(t)));

If you must use fsolve, try to restrict the domain.

fsolve({f, g}, {a, c}, a = 10000 .. infinity, c = 0 .. infinity);

Edit.
1. You may use
plots[implicitplot]([f,g], a=-20..20,c=-20..20,color=[red,blue]);
to localize the roots
2. The system seems to be intentionally constructed to have "non-intuitive" solutions.

Yes, obviously for the ics case Maple "forgot" the BesselJ term .

M := diff(T(r), r, r)+(diff(T(r), r))/r+u*T(r)+P*(r^4+r^2) = 0; # u includes the constant

M0 := {M, D(T)(0)=0}:
s:=dsolve(M);

s0:=dsolve(M0);


s0general:=T(r)=BesselJ(0, sqrt(u)*r)*_C2-P*(r^4*u^2+r^2*u^2-16*r^2*u-4*u+64)/u^3;

odetest(s, M);
                               0
odetest(s0, M0);
                              {0}
odetest(s0general, M0);
                              {0}

Edit. If another ics is added e.g.
M01 := {M, D(T)(0)=0, T(0)=a};
then BesselJ appears!

B:=Groebner[Basis]([x-v, y-v^2], plex(u,v,x,y)):
remove(has,B,[u,v]);