Let \({\mathcal X}(x,y)=p(x,y){{\partial}\over{\partial x}}+q(x,y){{\partial}\over{\partial y}}\) be a planar polynomial (real or complex) vector field of degree \(m,\) having an invariant algebraic solution of degree \(n,\) \(f(x,y)=0,\) i.e. \({\mathcal X}(f)=k f\) for some polynomial \(k(x,y)\) of degree \(m-1\) called the cofactor of \(f.\) Denote by \(P(X,Y,Z)= Z^mp(X/Z,Y/Z),Q,F\) and \(K\) the homogeneizations of the polynomials \(p,q,f\) and \(k,\) respectively. The aim of the paper is to get conditions on \({\mathcal X}\) to have a rational first integral, as well as to get upper bounds on \(n\) in terms of \(m\) and the number and type of singular points of the algebraic invariant curve. The reviewer describes some results with more detail: If it is assumed that the curves \(nP-XK=0, nQ-YK=0, KZ=0\) do not have any common component and that the curve \(F\) has finitely many multiple points in \({\mathbf C}P^2\) taking into account their multiplicities, namely \(h,\) then \((n-1)(n-m-1)\leq h.\) Notice that as a consequence of this result is the following if \(f(x,y)=0\) is an irreducible invariant algebraic curve without multiple points \((h=0),\) then its degree \(n\leq m+1.\) It is also proved that if \(n=m+1, \) \(m>1\) and \(F\) has no multiple points then \({\mathcal X}\) has a rational first integral. Finally, in the case in which all the multiple points of \(F\) are double and ordinary it is proved that \(n\leq 2m\) and moreover that \({\mathcal X}\) has also a rational first integral when \(n=2m.\)