R={{1,-1,1},{1,1,-1},{2,-1,0}}; MatrixForm[R]

x[t]={x1[t],x2[t],x3[t]};

gensol=DSolve[D[x[t],t]==R.x[t],x[t],t]

y[t_]=Simplify[{gensol[[1,1,2]],gensol[[1,2,2]],gensol[[1,3,2]]}]

Simplify[D[y[t],t]-R.y[t]]

eigenvalues=Solve[Det[R-r* IdentityMatrix[3]]==0,r]

r1=eigenvalues[[1,1,2]]

v={v1,v2,v3};

eigenvectors=Solve[R.v==r1*v,v]

vr1=v/.eigenvectors[[1]]

vr1=vr1/.{v1->1}

eigensystem=Eigensystem[R]

val=Eigenvalues[R]

vect=Eigenvectors[R]

X[t_]=Transpose[{Exp[val[[1]]*t]*vect[[1]],Exp[val[[2]]*t]*vect[[2]],Exp[val[[3]]*t]*vect[[3]]}];MatrixForm[X[t]]

Fundam[t_]=X[t].Inverse[X[0]];MatrixForm[Fundam[t]]

Simplify[D[Fundam[t],t]-R.Fundam[t]]

Det[Fundam[t]]

Simplify[Fundam[t]-MatrixExp[t*R]]

Z[t_]=Fundam[t].{C[1],C[2],C[3]}

Simplify[Z[t]-y[t]]

sol=Flatten[Table[Z[t]/.{C[1]->i,C[2]->j,C[3]->k},{i,-1,1,.5},{j,-1,1,.5},{k,-1,1,.5}],2];

ParametricPlot3D[sol,{t,-1,1}]