We now have a solution to a boundary value problem whose boundary conditions on the second brane are not yet known. We parameterize the brane potential on the second brane λ$_{2}$ ( χ$_{a}$ ) in terms of P parameters α$_{b}$ (for example, the brane tension Σ$_{2}$, the brane mass term m 2 a of each scalar, the quartic coupling of each scalar, etc.) Then the junction conditions (35)-(36) at the second brane ( i = 2) give N linear equations for the P unknowns α$_{b}$ . By evaluating the fields on the second brane, and using the param- eterization in (37), we then invert the N junction conditions to determine the α$_{b}$ . If this is possible, then the solution to our initial value problem is also a solution to a corresponding boundary value problem. From this we see that we must have P ≥ N in order to guarantee that the field configuration we obtained is the solution to a corresponding boundary value problem. If P = N , the α$_{b}$ are uniquely determined, and there is a unique Lagrangian for which the above field configuration is a solution. On the other hand if P > N , some of the α$_{b}$ are arbitrary and so there is a family of solutions for these final-boundary conditions. In that case there is a family of Lagrangians which yield the obtained field configuration, and one can proceed by focusing on one member of this family. If P < N , the linear system of parameters α$_{b}$ may be overdetermined, in which case the obtained field configuration is not a solution to any corresponding boundary value problem. We can find additional solutions by changing the initial-boundary conditions and repeat- ing the above process. Note that by freely varying the field derivatives ( φ $^{′}$, χ ′ 1 . . . χ ′ N − $_{1}$) at the initial brane and determining the remaining quantities from the junction conditions, it is possible to leave the initial-brane potential unchanged. This is necessary in order that the action remains unchanged (it is not sufficient because part of the action is determined by the final-brane potential). A solution and the resulting final-boundary conditions (the α$_{b}$ ) are then found as before. Since each set of initial shooting values yields a set of α$_{b}$ , each α$_{b}$ is a function of the N initial-boundary derivatives. Each α$_{b}$ therefore defines an N -dimensional surface whose level-surfaces can be projected onto the φ $^{′}$( y$_{1}$ )- χ ′ $_{a}$( y$_{1}$ ) parameter space (which is an N -dimensional space). In the above construction there are P such quantities, and so P level-surfaces intersect at every point in this parameter space, representing one solution for this action. The question of whether multiple solutions exist for the same action is equiva- lent to the question of whether the same P surfaces simultaneously intersect at more than 15 λ$_{2}$ ( χ$_{a}$ ) = f (Σ$_{2}$ , m 2 $_{1}$, m 2 $_{2}$, ..., m 2 $_{N}$, ... ) . (37)